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Full text of "Evolution of Particle Physics"

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PRESTON POLYTECHNIC 
LIBRARY & LEARNING RESOURCES SERVICE 

This book must be returned on or before the date last stamped 




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sfrti 



y 



Evolution 



of 



Particle Physics 



A volume dedicated 

to 

Edoardo Amaldi 

in his sixtieth birthday 



BRITISH LIBRARY 
Edited by M. Conve^si LE 

16 JAN 1978 



I 



Y 2 7469 










ACADEMIC PRESS • new york and london 



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New York 3, N, Y. 



United Kingdom Edition 

Published by 

ACADEMIC PRESS INC. (London) Ltd. 

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Copyright © 1070, ay Academic Press Inc. 

ALL RIGHTS RESERVED 

NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, 

BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, 

WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. 

Library of Congress Catalog Card Number: 70- J 17640 



ACCESSION No. 



128830 



CLASS No. 




12 OCT 1978 J 



H 



PRINTED IN ITALY 



To Edoardo Amaldi 
in his sixtieth birthday 



Biographical Note 



Edoardo Amaldi was born in Carpeneto Piacentino, Piacenza, on Sep- 
tember 5, 1908. He graduated from the University of Rome in 1929. In the 
years 1931-36 he consolidated his scientific formation at the University of 
Lipsia under P. Deoye, the Cavendish Laboratory at Cambridge directed 
by Lord Rutherford, Columbia University in New York, and the Carnegie 
Institute in Washington. He has been professor of experimental physics at 
the University of Rome since 1937. 

After a forced absence due to the second world war, Amaldi is again 
in Rome at the end of the Italian participation to the conflict. He was Director 
of the former Center of Nuclear Physics of the Italian C.N.R, from 1945 
to 1952, Vice-President of U1PPA from 1948 to 1954, Director of the Physics 
Department of the University of Rome from 1949 to I960, Secretary General 
of CERN from 1952 to 1954, Director of the Roman Section of INFN from 
1952 to 1960, Director of the « Scuola di Pcrfezionamento in Fisica» in 
Rome from 1952 to 1966, Vice- Director of CERN in 1954 and 1955, Vice- 
President of the Italian CNRN, later CNEN, from 1956 to 1960, Pres- 
ident of UIPPA from 1957 to I960, Chairman and later Member of the 
EURATOM Scientific and Technical Committc from 1958 to 1961, Chairman, 
from 1958 to I960, and Member, from I960 to the end of 1969, of the CERN 
Scientific Policy Committee, President of INFN and Member of the CNEN 
Directive Committee from I960 to 1965, Vice-President of INFN from 1965 
to 1968, Chairman of the European Committee for Future Accelerators 
(ECFA) from 1963 to the beginning of 1970, when he became President of 
the CERN Council. 

Amaldi is a member of the «Accademia Nazionale dei Lincei », the 
« Accademia Nazionale dei XL», the « Accademia Pcloritana», the « Acca- 
demia delle Scienze di Torino », the « Istituto Lombard© », the « Accademia 
Nazionale di Scienze, Lettere ed Art! di Modena», the « Accademia Pu- 
gliese detle Scienze », the « Istituto Nazionale delle Profession!, delle Arti 
e delle Scienze », the « Accademia Leopoldina», the Royal Society, the 
Royal Society of Sciences of Uppsala, the Academy of Sciences of the USSR, 
the Royal Institution of Great Britain, the American Philosophical Society 



VI Biographical note 

of Philadelphia, the National Academy of Sciences, the American Academy 

of Arts and Sciences of Boston, the International Academy in Astronautics, 
the « Academic Royalc Neerlandaise ». 

Amaldi has received a number of prizes: one from the International 
Columbus Association, also the « Premio Ibbico-Reggino », « Premio Righi », 
«Premio Sella », « Premio Uranio »... He is doctor honoris causa from the 
University of Algiers. He has been awarded the Seal of the Sorbonne and, 
recently, the golden medal of the Italian Physical Society, 



Contents 



Biographical Note , P a §- v 

Presentazione . , , , ,,.».. » K 

Foreword , >J Xlt 

Edoardo Amafdi et le CERN . . . . » *V 

.d Edoardo Amaldi » XDE 

7b Edoardo Amaldi, on his sixtieth birthday » xxi 

Scientific Contributions » xxm 

L. W. Alvarez - Recent developments in particle physics ...» 1 
N. Cabibbo and L. Maiani - Weak interactions and the breaking 

of hadron symmetries » 50 

G. Cocconi - The role of complexity in nature » 81 

B. Ferretti - Channeling of ullrarelativislic charged particles in 
crystals » 88 

G. Fidecaro and M. Fidecaro - Experimental work on coherent 

scattering of high-energy hadrons by light nuclei » 95 

C. Fkan/lnliii and G, Stoppini - The determination of the axial 

vector coupling for strangeness nonchanging currents .... » 114 

S. Fubini - Old problems and new ideas in elementary particle 

physics ■ }> '-'-' 

R, Gatto - High energy e"-e~ annihilation into hadrons .... » 138 

G. Giacomelli - New frontiers of high-energy physics » 148 

M. Goldhaber and G N. Yang - The K°-K> system in p-p anni- 
hilation at rest ............-■■ » l71 

T. D. Lee - Symmetry principles in physics .......... » 176 

A. W. M urrison - The design and use of large electron synchrotrons » 1 84 



VIH Contents 

L. Michel and L. A, Radicati - Breaking of the SU 3 x$U 3 sym- 
metry in hadronic physics , pag. 191 

A. Paoletti and S. Sciuti - Structure of matter investigations by 
thermal neutrons in Rome » 204 

B. Pcxntecorvo - Search for new stable particles ....... » 210 

R. A. Rtcci - The isobaric analog resonances in phenomeno logical 

nuclear spectroscopy » 21S 

B. Rossi - The Crab nebula. Ancient history and recent discoveries » 237 

C. Rubbia - The K L — K s mass diiTercnce » 257 

J. Steinberger - Suggestion for a more precise measurement of 

the %_ phase » 268 

V. F. Weisskopf - An amateur's view of particle physics .... » 273 

G. Wick - Some questions concerning adiabatic transformations » 287 

R. R. Wilson - Range and straggling of muons » 294 

A. Zichichi - The Basic SU a mixing: ^±7% ........ » 299 

List of Papers by Edoardo Amaldi » 335 



Presentazione 



Circa un anno fa, il 5 settembre del 1968, Edoardo Amaldi ha varcato il 
limite simbolico della matura eta. Eravamo con lui, in quelVoccasione, in un 
piccolo gruppo di amid seduti intorno al tavolo di un modesto ristorante Vien- 
nese, durante una breve pausa della International Conference on High Energy 
Physics. Non vifu alcun festeggiamento. Ma al termine della colazione un breve 
scambio di battute tra alcuni di noi segno la ripresa di una iniziativa che era 
stata interrotta da motivi di carattere contingente. Quell 'iniziativa si conclude 
ora con la pubblicazione del presente volume, il quale — tardivamente ma non 
meno affettuosamente—vuole appunto commemorare il 60° compleanno di 
Edoardo Amaldi. 

II titolo Evolution of Particle Physics sembra appropriato ai quaranta anni 
di vita da fisico vissuti dalVuomo cui il libro e dedicato. La vita scientifica di 
Edoardo Amaldi si svolge infatti su un arco di secolo che ha visto nascere ed 
evolversi fino aWattuale livello di complessita Vintero campo della fisica delle 
particelle. Prima degli anni 30 si conoscevano solo due particelle, ritenute i 
costituenti fondamentali di tutto VUniverso. A parte la scoperta di nuove e 
nuove particelle {che non fa piii senso chiamare oggi « elementari ») il concetto 
di costituente fondamentale si e trasferito in quelVarco di secolo dal mondo 
del nucleo atomico a quello degli stessi corpuscoli di cui esso e formato, attra- 
verso modelli ed ipotesi come quella — la piu recente — associata al nome dei 
« quarks ». QuelVarco di secolo ha visto mutare per molti ordini di grandezza 
— dai milioni ai miliardi di elettronvolt — le energie tipiche dei fenomeni inve- 
stigati edha vissutofin dalla nascita tutte lefasi dello sviluppo degli acceleratori: 
dalla prima macchina elettrostatica operante nella regione dei cento keV, fino 
ai giganteschi acceleratori per protoni da centinaia di GeV, ora in costruzione 
o in progetto. Uevoluzione di quel quaranfanni coinvolge la fisica teorica cosi 
come le tecniche sperimentali. Sorge e si sviluppa il nuovo campo delle intera- 
zioni deboli, Videa del mesone come agente delle forze nucleari, la nuova elettro- 
dinamica quantistica, la teoria dei campi nella sua versione moderna; mentre 
dalla camera a nebbia e dai contatori proporzionali usati con una elettronica 
primitiva, si passa alle gigantesche camere a bolle di oggi ed ai complessi sistemi 
elettronici e di camere a traccia impulsate elettricamente, caratteristici degli 
odierni esperimenti presso i grandi acceleratori di particelle. 



X Presentazione 

Alia scena di questa evoluzione — cut il nostro paese e VEuropa hanno dato 
ben tangibili contributi — Edoardo Amaldi ha partecipato come « at tore » 
sempre presente. Vi ha partecipato sia come scienziato avente al suo attivo una 
vastissima gamma di ricerche, sia assumendo a livello nazionale ed europeo 
pesanti e numerose responsabilitd direttive. Perche prima ancora che un grande 
organizzatore egli e stato ed e un uomo di scienza: un fisico. Nel suo primo 
lavoro sperimentale, compiuto da studente non ancora ventenne, e Vinizio di 
un'attivitd di ricerca, sempre viva e di frontiera, che si protrarra ininterrotta- 
mente per i successivi quaranfanni, nonostante i pesanti impegni connessi alle 
cariche direttive sopra accennate. 

Nei primi anni la ricerca si svolge in seno aWormai famoso gruppo dei 
« ragazzi di Corbino » dei quali Edoardo e allora il piii giovane. II gruppo, 
formato da giovani di primissimo ordine e dominato daW eccezionale personalita 
scientifica di Enrico Fermi, e destinato a disfarsi nel giro di un decennio, sotto 
Vincalzare degli eventi che preludono alio scoppio della seconda guerra mondiale. 
Ma in questi died anni si compiono lavori destinati a restare nella storia della 
fisica. Amaldi partecipa attivamente alle ricerche sperimentali, le quali spa- 
ziano dalla fisica atomica, alia spettroscopia molecolare, fino ai classici esperi- 
menti sulV interazione dei neutroni con la materia. Tra i risultati piii salienti 
di questi esperimenti sono la radioattivitd indotta dai neutroni, con la produ- 
zione di numerosi nuovi radioisotopi, la scoperta del rallentamento neutronico, 
la determinazione delle piii importanti proprietd dei neutroni lenti con la prima 
verifica sperimentale del teorema ottico, la prova diretta della diffrazione dei 
neutroni veloci da parte dei nuclei. 

Ogni possibilitd di ricerca e virtualmente bloccata negli anni tragici della 
guerra. Gid prima, con la scomparsa prematura di Ettore Majorana e poi con 
Vesodo di quasi tutti i componenti della Scuola di Roma, questa minaccia di 
estinguersi. E Amaldi a salvarla, con la sua presenza costante, particolarmente 
nei momenti dijficili, con la sua dedizione assoluta, con il suo esempio, con la 
volontd di riprendere Vattivita di ricerca non appena Voccasione se ne presenta. 
E la sua attivita scientifica negli anni del dopoguerra si orienta verso lo studio 
dei raggi cosmici prima, poi delle particelle elementari. Con tecniche elettro- 
niche che preludono alia futura complessitd degli apparati usati presso gli acce- 
leratori di alta energia, investiga le proprietd dei mesoni cosmici a grandi pro- 
fondita. Con la tecnica delle emulsioni nucleari esposte ad alta quota mediante 
palloni sonda, studia il comportamento dei pioni carichi, dei mesoni K e degli 
iperoni presenti nella radiazione cosmica ed osserva un probabilissimo esemplare 
di stella da annichilazione antiprotone-nucleone precedendo la scoperta conclusi- 
va delV antiprotone compiuta a Berkeley nel 1955. Poi dal 1955 al 1959 guida il 
gruppo di Roma che insieme ad un gruppo di Berkeley compie le prime ricerche 
sistematiche suW annichilazione degli antiprotoni prodotti con il Bevatron. 



Presentazione XI 

Negli anni successivi troviamo Amaldi — ormai piii che cinquantenne — 
ancora in prima fila, coinvolto in arditi esperimenti presso i grandi accelerator! 
diparticelle: dalla ricerca dei monopoli di Dirac, alio studio della polarizzazione 
dei protoni di rinculo nelVurto elettrone-protone, fino alle recentissime espe- 
rienze di elettroproduzione. 

II contributo di Amaldi come organizzatore della fisica in Italia e in Europa 
e stato poi veramente eccezionale. Questo volume inizia con uno scritto di 
Francis Perrin che illustra il contributo particolare apportato da Amaldi alia 
costituzione e agli sviluppi del CERN. Con analogo entusiasmo e con la stessa 
dedizione, egli disimpegnava in queWepoca le altre pesanti cariche direttive in 
seno alVUniversita di Roma, alVINFN, al CNEN, ecc. : mai trascurando 
V attivita didattica; mai tralasciando di seguire da vicino le esperienze in cui 
era coinvolto, ne gli sviluppi delle ricerche anche in settori lontani da quello del 
suo immediato interesse; promuovendo anzi la costituzione di nuovi gruppi di 
ricerca a Roma nei campi delle basse temperature e della fisica spaziale; mai 
rifiutandosi di sacrificare la sua attivita personate a favore di iniziative — come 
ad esempio il « Pugwash » — che dalV autorita scientifica della sua persona 
potessero trarre vantaggi per le generazioni a venire. E questa capacita di disim- 
pegnare con continuita, ininterrottamente sopra un arco di decenni, una cost 
varia e complessa attivita, appare quasi prodigiosa alia luce dello stile sobrio, 
semplice, disinvolto, sempre limpido e sereno, tipico di Amaldi. Lo ricordo 
ancora negli anni piii oscuri deW occupazione nazista — quando piii o meno 
nascosti nel seminterrato di un liceo romano conducevamo lontani dalVUni- 
versita bombardata i nostri esperimenti sui mesoni cosmici — incontrarsi spesso 
con noi, sempre esattamente aWora prefissata per V appuntamento, come se 
intorno tutto fosse normale; e sempre recando, insieme al contributo della sua 
cultura e della sua sensibilita difisico, il sostegno — in quel giorni ancora piii 
prezioso — di una fondamentale fiducia nelVavvenire. 

Nel quarto di secolo trascorso da quel ricordo, passato il traguardo dei 
sessanf anni, ritroviamo in lui quella stessa fondamentale fiducia. Essa gli ha 
permesso di superare in tutta serenitd le gravi crisi che in quel periodo hanno 
travagliato la ricerca scientifica e la scuola italiana; e poiche e parte integrante 
della sua stessa personalitd, essa gli sard sempre accanto nei lunghi anni a venire. 
Noi suoi amici — anche i tanti che non hanno potuto manifestargli la low stima 
e la loro simpatia con uno scritto su questo libro — esprimiamo il fervido augurio 
che la scuola italiana, la ricerca scientifica europa e la fisica nel suo complesso, 
continuino per molti e molti anni a ricevere, come sempre in passato, i frutti 
della sua instancabile operositd e della sua equilibrata saggezza. 

Marcello Conversi 
Roma, Ottobre 1969 



Foreword 



About one year ago, the 5th of September, 1968, Edoardo Amaldi cele- 
brated his sixtieth birthday. On that occasion we were with him, a small group 
of friends, sitting in a modest restaurant in old Vienna, during a break in the 
1968 Rochester International Conference on High Energy Physics. There 
was no ceremony. But after lunch, a few words exchanged between some of us 
marked the resumption of an enterprise interrupted by the exigencies of our 
professional lives. That enterprise is now concluded with the publication of 
the present volume, which tardily, but no less affectionately, commemorates 
Amaldi' s sixtieth birthday. 

The title Evolution of Particle Physics seems very appropriate to the man 
to whom the book is dedicated, with his scientific life covering the entire period 
from the birth of the field of particle physics to its development to the present 
level of complexity. Before 1930 only two particles were known and believed 
to be the fundamental bricks of the whole universe. Since then, not only have 
many new particles been discovered — which can no longer logically be called 
« elementary » — but even the idea of a few fundamental bricks has moved 
from the realm of the atomic nucleus to that of its own consistuents through 
hypotheses and models such as that of the « quarks » {the most recent one). 
The energies involved in the phenomena investigated have changed from the 
million to the billion electronvolt region; while the first particle accelerator, 
an electrostatic machine operating at some tenths of a mega electronvolt, 
evolved to the giant machines now under construction or in project, for protons 
of several hundred giga electronvolts. The evolution of the last forty years 
involves theoretical physics as well as experimental techniques. The new field 
of weak interactions, the idea of the meson as the agent of nuclear forces, the 
new quantum electrodynamics, the modern version of field theory, all evolved 
in these forty years; whereas from the cloud chamber and the proportional 
counter used with a primitive electronics, one has ended with the huge bubble 
chambers and the gigantic electrically pulsed track detectors, associated with 
complex electronics, now currently used in high-energy physics. 

Amaldi was always present as an « actor » on the stage of this evolution, 
to which our country and Europe have given an appreciable contribution. He 



Foreword XJII 

has been there both as a scientist with a vast spectrum of interests and as an 
organizer of Italian and European scientific research. Notwithstanding that he 
is a great organizer, he is first and foremost a physicist. 

His first work, carried out when he was a nineteen-year-old student, was 
the beginning of a scientific career always on the frontier of physics, lasting 
uninterruptedly for over forty years. At the beginning, his research was car- 
ried out in the now famous group of the « Corbino's boys », where Edoardo 
was the youngest. The group, dominated by the scientific personality of Enrico 
Fermi, included a number of first class young scientists. Hounded by the events 
preceding the outbreak of the second world war, the group was destined to 
disperse in about a decade. But in these ten years work was accomplished which 
remains in the history of physics. Amaldi actively participated in all the experi- 
mental research, which ranged from atomic physics to molecular spectroscopy, 
up to the classic experiments on the interaction of neutrons with matter. Among 
the most significant results are the radioactivity induced by neutrons, the pro- 
duction of new radioisotopes, the discovery of the slowing of neutrons, the first 
experimental check of the optical theorem, the direct proof of the diffraction 
of fast neutrons by nuclei. 

Any possibility of research was virtually eliminated in the tragic years of 
the war. Even before, after the premature disappearence of Ettore Majorana 
and the exodus of almost all components of the Roman school, the latter was 
near extinction. Amaldi saved it by his continuous presence, especially in difficult 
moments, by his total dedication, his example, his will to recommence scientific 
activity as soon as circumstances would allow. In the post-war years his interest 
moved first to the field of cosmic rays and later to elementary particle physics. 
He investigated the properties of cosmic ray muons underground, using an 
electronic technique which prefigured the future complexity of the apparatuses 
later employed with high-energy accelerators. Then he studied the behaviour 
of cosmic ray pions, K-mesons, and hyperons, using nuclear emulsions exposed 
in balloon fligths, and observed a very probable example of an antiproton- 
nucleon annihilation star, just before the conclusive discovery of the antiproton 
at Berkeley in 1955. Subsequently, from 1955 to 1959, he led the Roman group 
which together with a group at Berkeley performed the first systematic inves- 
tigation of the annihilation of the antiprotons produced by the Bevatron. 

In the following years we find Amaldi — now more than fifty years old — still 
in the first line, involved in bold experiments carried out with large particle 
accelerators. They include a search for Dirac monopoles, a study of the polar- 
ization of recoil protons in electron-proton collisions, and the very recent experi- 
ments on pion electroproduction. 

The contribution of Amaldi to the organization and development of physics 
in Italy and Europe has been really outstanding. This book, starts with a paper 



XIV Foreword 

by Francis Perrin which illustrates particularly AmaldVs contribution to the 
creation and development of CERN. With the same enthusiasm and dedication, 
he performed in that period the heavy duties associated with the directive respon- 
sibilities he had at the University of Rome, the Italian Institute of Nuclear 
Physics (INFN), the National Committee for Nuclear Research (CNRN, 
later CNEN), etc.; never neglecting his educational duties, never failing to follow 
closely the experiments in which he was personally involved, nor the develop- 
ments of research also in fields far from his immediate interest, promoting, 
in fact, the creation of new groups in Rome operating in low temperature and 
space physics, never refusing to sacrifice his personal activity in favour of ini- 
tiatives — such as, for example, « Pugwash » — which through the weight of his 
scientific authority could bring advantages to the generations to come. 

The capacity to engage in activities so various and complex, continuously 
and uniterruptedly over a period of decades, appears almost prodigious in the 
light of AmaldVs personality, sober, informal, easy going, always clear minded 
and serene. I still remember him in the darkest years of the Nazi occupation 
(when more or less hidden in a Roman lyceum, we carried out far from the 
bombarded University our experiments on the cosmic muons) meeting often 
with us, always exactly at the prefixed time, as if everything around were 
normal. And always bringing, together with his culture and physical good sense, 
the support — even more precious in those days — of a fundamental confidence 
in the future. 

A quarter of a century later, we still find in him the same fundamental 
confidence. It has allowed him to overcome the severe crisis that scientific 
research and the Italian school suffered in that period. And since this confi- 
dence is an integral part of his personality, it will always accompany him in 
the long years to come. His friends — including the many friends who were 
unable to exhibit their esteem and affection with a contribution to this book — 
express with me the warmest wish that the Italian school, European scientific 
research and physics at large, will continue for many and many years to come 
to profit, as in the past, from the fruits of his tireless work and serene wisdom. 

Marcello Conversi 

Rome, October 1969 



Edoardo Amaldi et le CERN 



Depuis une trentaine d'annees les progres les plus significatifs de la phy- 
sique fondamentale se sont produits dans le domaine des particules ephe- 
meres creees par des collisions entre nucleons a des energies de plus en plus 
elevees. Tant que les rayons cosmique on ete la seule source de particules 
de haute energie pouvant donner lieu a de telles collisions, les physiciens 
europeens travaillant en Europe, avec les techniques des emulsions nucleaires 
ou des chambres a brouillard, ont eu une grande part dans les decouvertes 
de particules nouvelles qui ont fait pressentir la richesse imprevue et l'etran- 
gete de ce domaine ou un nouveau bouleversement des fondaments de la 
Science commencait a s'esquisser. 

Mais, des la mise en service des accelerateurs americains de quelques 
gigaelectron-volts, le Cosmotron de Brookhaven puis le Bevatron de Ber- 
keley, il est clairement apparu aux physiciens les plus avertis que si de grands 
accelerateurs n'etaient pas construits en Europe, les scientifiques europeens 
ne pourraient plus participer effectivement au progres de cette partie de la 
Science, celle ou des lois vraiment nouvelles pouvaient etre decouvertes, 
sans emigrer vers l'Amerique, ce qui ferait peu a peu perdre a la grande 
majorite des nouvelles generations d'etudiants euroeens tout contact vivant 
avec la Science en formation. 

Edoardo Amaldi fut un des quelques hommes qui eurent tres tot et tres 
vivement conscience de ce danger, et qui, comprenant que la construction 
d'un accelerateur equivalent aux futurs accelerateurs americains exigerait des 
moyens financiers et humains beaucoup plus grands que ceux qu'aucun Etat 
europeen, agissant isolement, pourrait ou voudrait consacrer a une telle rea- 
lisation, entreprirent une action visant a associer la plupart des pays d'Europe 
en vue de la construction en commun d'un accelerateur des tres haute energie. 

Cette action, trouvant heuresement un climat politique favorable, devait 
aboutir relativement rapidement a la signature en 1953, par une douzaine 
d'Etats europeens, d'une convention creant un Centre Europeen de Recherche 
Nucleaire, le CERN, etabli a Geneve par un choix unanime, qui avait 
comme object principal la construction et 1' exploitation scientifique d'un grand 
accelerateur. Le remarquable succes de cette entreprise, tant pendant la 
phase de la construction que pendant celle de 1' exploitation, succes qui a 



XVI Edoardo Amaldi et le CERN 

permis a l'Europe de reprendre, apres une dizaine d'annees d'eclipse, une 
place de premier rang en physique des hautes energies, est certainement du 
au fait qu'elle a ete concue puis animee par des hommes de science qu ne 
perdaient jamais de vue le but poursuivi et qui faisaient aisement abstraction 
de toute rivalite nationale mesquine. 

Les deux physiciens qui ont par leur action le plus contribue a la creation 
du CERN sont certainement Pierre Auger et Edoardo Amaldi. C'est a la 
suite d'entretiens qu'il eut avec l'un et l'autre, que Rabi chef de la delegation 
americaine a l'Assemblee generate de l'UNESCO qui se tint a Florence au 
printemps de 1950, fit adopter une resolution chargeant cette organisation 
mondiale de susciter la creation d'un laboratoire europeen de physique des 
hautes energies. Pierre Auger alors Directeur du Departement des Sciences 
exactes et naturelles de l'UNESCO, prit en main la mise en oeuvre de cette 
resolution en faisant etablir un projet precis, qui puisse etre propose aux 
gouvernements, par un groupe d'experts, physiciens des principaux pays 
interesses choisis en accord avec Amaldi. Des la premieere seance de ce 
groupe d'expert, Auger en confia la presidence a Amaldi, reconnu par tous 
comme le plus qualifie, pour en animer les travaux. 

Quand en mai 1952 une Organisation provisoire du futur CERN fut 
decidee par un accord entre les gouvernements, Amaldi en fut nomme Secre- 
taire General. A ce poste, qu'il conserva jusqu'en octobre 1954, il dirigea la 
preparation de la convention definitive, avec ses annexes fmancieres, et les 
reglements qui donnerent au CERN sa grande efficacite de fonctionnement. 
Durant cette periode de tres important contrats furent prepares et signes 
avec l'architecte et les principales compagnies qui devaient construire ou 
fabriquer les batiments et les premiers gros constituants (dont l'aimant) du 
premier accelerateur, le synchrocyclotron de 600 MeV, dont devait etre 
dote le CERN. De nombreux contrats relatifs a la construction du synchro- 
tron a protons de 25 GeV, appareil principal du futur laboratoire, furent 
egalement prepares sous la direction d'Amaldi. 

Apres la nomination, en octobre 1954, de Felix Bloch comme premier 
Directeur General du CERN, Amaldi accepta encore de consacrer la plus 
grande partie de son activite au CERN pendant 6 mois en qualite de Directeur 
General adjoint. Pendant les annees cruciales de formation et d'orientation 
du CERN, durant lesquelles s'est forme l'esprit qui devait l'animer quand la 
recherche scientifique a pu y commencer, c'est Amaldi qui a joue avec con- 
tinuity le role principal, en montrant un grand sens de Taction politique, mais 
en pensat et en agissant toujours en grand physicien, qui ne perdait jamais 
de vue l'objectif scientifique de cette vaste entreprise commune europeenne. 

Malgre tant de temps et d'effort donnes a des activites administrative, 
Amaldi a su rester un vrai scientifique et aussitot qu'il l'a pu il s'est de nouveau 
consacre a ses activites universitaires a Rome, enseignement et direction d'un 
Institut de Recherche tres vivant. Et, ce qui est particulierement remarquable, 



Edoardo Amaldi et le CERN XVII 

il a su reprende lui-meme une activite originale de recherche. C'est ainsi 
qu'il a effectivement participe, au cours d'un sejour a Berkeley, a la tres 
belle experience qui apporta, en octobre 1956, la preuve definitive de l'exi- 
stence des antiprotons par l'observation et l'analyse precise d'etoiles produites 
par leur annihilation dans des emulsions nucleaires. 

C'est ce qui permet de comprende pourquoi, quand au debut de 1963 
il apparu necessaire d'envisager pour l'Europe, en face de project americains 
activement pousses, une deuxieeme etape de collaboration en physique des 
hautes energies, c'est encore Amaldi que le Comite des Directives Scientifi- 
ques du CERN designa a l'unanimite pour presider un Comite europeen 
sur les futurs Accelerateurs, l'ECFA, pourtant deliberement forme par des 
physiciens en pleine activite de recherche et nettement plus jeunes que ceux 
avaient prepare la fondation du premier CERN. Le rapport presente par 
Amaldi des le mois de juin 1963 recommandait la construction, en comple- 
ment du PS de 28 GeV, d'anneaux de stockgae a intersections (les ISR) 
devant permettre une exploration a tres faible intensite vers les tres hautes 
energies, et la creation d'un nouveau laboratoire ayant comme equipement 
principal un synchrotron a protons de 300 GeV (environ dix fois plus grand 
que celui du premier laboratoire europeen). La construction des ISR fut 
decidee par le Conseil du CERN en juin 1965 et doit s'achever en 1971, mais 
malgre un rapport technique detaille presente en 1964 par l'ECFA, aucune 
decision ne fut prise relativement a l'accelerateur de 300 GeV. Afin de refaire 
un examen approfondi des besoins europeens et des possibilites techniques 
de realisation dans le domaine des tres hautes energies, l'ECFA fur renouvele 
en mars 1966, groupant alors une soixantaine de physiciens, toujours sous la 
presidence d'Amaldi, qui presenta au Conseil du CERN en juin 1967 un 
nouveau rapport confirmant la recommandation de la realisation du synchro- 
tron de 300 GeV propose en 1964 et tracant les grandes lignes du programme 
de son utilisation scientifique. C'est a la suite de ce rapport que furent adres- 
sees au President du Conseil du CERN les premieres lettres officielles de 
gouvernements declarant leur intention de participer a un deuxieme labora- 
toire europeen dote d'un accelerateur de 300 GeV, si un nombre suffisant 
de partenaires pouvaient etre rassembles. 

La grande valeur des considerations generates justifiant les propositions 
presentees par Amaldi en tant que president de l'ECFA doit etre soulignee. 
Dans ses rapports au Comite des Directives Scientifiques ou au Conseil du 
CERN, on sent qu'il ne s'est jamais laisse influencer par des arguments de 
prestige ou de rivalite avec l'Amerique ou l'URSS, ni par le desir d'etablir 
des records ou l'attrait du gigantisme; ses raisons etaient toujours solidement 
fondees sur l'interet scientifique et une utilisation optimum, a ce point de 
vue, des grands moyens financiers inevitablement necessaires. C'est ainsi 
qu'il a clairement montre que le choix d'une energie de 300 GeV pour le 

2* 



XVIII Edoardo Amaldi et le CERN 

projet de futur accelerateur europeen, tandis que les Americains envisageaient 
alors pour leur projet une energie de 200 GeV, ne resultait pas du desir d'at- 
teindre une energie un peu plus grande, avec peut-etre, sans raison serieuse, 
l'espoir de depasser le seuil d'apparition d'un phenomene nouveau, mais 
que ce choix etait justifie par la prevision que Ton obtiendrait, dans le domaine 
d'energie autour de 10 GeV, des faisceaux secondaires de particules etranges 
(mesons K ou hyperons) et de neutrinos d'un ordre de grandeur plus intense 
avec un faisceau primaire de protons de 300 GeV au lieu de 200 GeV, et que 
c'etait le tres grand accroissement d'intensite de ces faisceaux secondaires 
d'energie relatievement basse qui permettait le plus d'esperer des progres 
decisifs dans la decouverte et la comprehension des lois fondamentales regis- 
sant le monde des particules elementaires. 

Ce sont d'ailleurs principalement ces raison s qui ont conduit les Ame- 
ricains a modifier leur projet initial en entreprenant la construction d'un 
accelerateur devant bien avoir initialement l'energie prevue de 200 GeV, 
mais pouvant etre, quelques annees apres, transforme, sans depense excessive, 
de facon a atteindre 400 GeV. Similairement, quand en juin 1968 le gouver- 
nement britannique fit savoir qu'il ne pourrait pas, pour des raisons finan- 
ciers, s'associer a la realisation du deuxieme laboratoire du CERN, malgre 
l'excellence du projet presente, et qu'il fallut pour sauver cette entreprise 
essentielle pour l'Europe, envisager un projet moins onereux, Amaldi, parlant 
toujours en tant que president de l'ECFA mais avec sa puissance person- 
nels de persuasion, reussit a convaincre tous les membre du Comite des 
Directives Scientifiques, que meme au prix d'autres sacrifices il fallait abso- 
lument garder la possibilite d'atteindre finalement l'energie de 300 GeV 
meme s'il apparaissait inevitable de proposer aux gouvernements une pre- 
miere etape ne depassant pas l'energie de 200 GeV. II ne pouvait accepter 
que des contingences sans doute passageres risquent de compromettre defi- 
nitivement ce qu'il voyait clairement etre la caracteristique la plus promet- 
teuse a longue echeance du futur accelerateur europeen. 

Ainsi, Edoardo Amaldi a ete depuis une vingtaine d'annees un des grands 
promoteurs du developpement de la physique des hautes energies en Europe, 
au premier rang de ceux qui estiment que la participation active a ce domaine 
de la recherche scientifique fondamentale est un facteur important pour 
conserver une ame a notre civilisation moderne menacee d'etouffement spi- 
rituel par l'aspect materiel du progres technique. II ne s'est jamais derobe 
quand on lui a demande de sacrifier ses activites personnelles de physicien 
car il avait conscience de pouvoir aider a preparer, pour toute une generation 
de jeunes chercheurs de nos vieux pays d'Europe, la possibilite de contribuer 
a certains des progres les plus significatifs de la Science. 

Francis Perrin 



A Edoardo Amaldi 



Piuttosto che strizzarmi il cervello per scrivere una pagina di fisica piu 
o meno interessante preferisco offrire a questo libro e a te personalmente 
una pagina di ricordi di un'amicizia di oltre quaranta anni. 

II prologo si stende su due o tre generazioni, originanti da Bozzolo. 
Mio padre raccontava che da bambino, ormai circa un secolo fa, gli dice- 
vano per farlo star buono che se era cattivo lo avrebbero detto al Giudice 
Amaldi, tuo Nonno. I nostri zii furono compagni al Collegio Ghislieri a Pavia 
e ancor prima di conoscerti ho incontrato i tuoi zii e cugine e ascoltato un 
corso sulla teoria delle funzioni di variabile complessa dato da tuo Padre. 
Era cosi bello che malgrado non fosse obbligatorio e fosse a un'ora scomo- 
dissima non ne persi una lezione. 

Noi ci siamo conosciuti verso il 1925 e andammo subito ad arrampicare 
montagne insieme. Non dubito avrai ormai digerito, ma non dimenticato, 
una scatoletta di antipasto sul Corno Piccolo. Allora — anche adesso — eri 
roseo, atletico e bellino. Quando diventammo compagni per qualche corso 
universitario ricorderai l'incarico di fiducia che avevi alle lezioni di Volterra, 
oltre a quello di prendere gli ottimi appunti su cui imparammo vuoi il doppio 
delta due che le distorsioni. Ci si preparava poi agli esami nella casa deimiei 
genitori a Tivoli; una cosa semplice rispetto a quello che mi dicono richiedi 
oggi ai tuoi studenti. 

Lo studio con Fermi dal 1927 in poi e per tutti quelli che hanno avuta la 
fortuna di parteciparvi un'esperienza indimenticabile, e se non bastasse 
aveva come soggetto principale l'allora nuova meccanica quantistica. Fu que- 
sto certo il punto saliente della nostra formazione culturale come fisici. 
Allora, a Via Panisperna, avemmo anche le visite dei nostri amici stranieri, 
soprattutto l'indimenticabile Placzek e ogni tanto Majorana che ci strabiliava 
coi suoi gruppi. Ricorremmo a tuo Padre perche la mia esperienza di alunno 
mi aveva persuaso che se c'era una speranza di imparare la teoria dei gruppi, 
era quella di farsela insegnare da lui. In risposta alle nostre richieste piut- 
tosto utilitarie, da fisici, egli ci disse che veramente si sarebbero dovuti studiare 
anche i gruppi di Lie. Era profeta e si rallegrerebbe oggi a vedere VSU 3 e i 
fisici curvi a studiarli. 



XX A Edoardo Amaldi 

Del lavoro a Roma voglio solo ricordare le docce di mercurio che pren- 
demmo al tempo degli atomi gonfi e la lampada per l'ultravioletto nello stesso 
periodo, nonche la tua preziosa lastra Raman dell'ammoniaca ; pero non 
tutti gli esperimenti avevano avventure cosi drammatiche, per fortuna. 

II lavoro sui neutroni con Fermi e ormai parte della storia maggiore della 
fisica. Dette occasione alia nostra breve permanenza a Cambridge, quando 
stava per nascere Ugo jr., e alia nostra conoscenza con Rutherford. Piu tardi, 
quando io ero ormai a Palermo, passammo Testate assieme in Dolomiti, 
con i figlioli piccini, in cesti, sui prati d'Alba di Canazei. 

Vennero poi gli anni tragici della guerra e delle persecuzioni. Eri a casa 
mia in California quando scoppio la seconda guerra mondiale e mi ricordo 
ancora le ansie di quel fatale settembre del 1939 e il tuo ritorno in Italia. 

Rotte le comunicazioni per la durata della guerra, le prime notizie che 
riebbi dall'Italia mi arrivarono a Los Alamos dove mi fu comunicato che i 
militari Americani ti avevano rintracciato, secondo ordini ricevuti, appena 
entrati a Roma. 

Ci rivedemmo nel 1947 quando tornai per la prima volta a Roma e i 
ricordi freschi della guerra erano il soggetto principale di conversazione. 
Appresi cosi le tue peripezie in Libia e poi la tua fabbrica di timbri e altre 
amenita del genere. 

Piu tardi, quando si cominciava a parlare del CERN per cui avevi tanto 
entusiasmo, se ne discusse a lungo, un'altra volta sotto i vecchi olivi diTivoli. 
Seguirono le tue nuove visite a Berkeley intercalate dalle mie a Roma. L'ul- 
tima volta che sei venuto a Berkeley fu al tempo del lavoro sull'antiprotone 
che ci ha dato occasione di scrivere di nuovo un lavoro insieme. 

A un tempo dove le amicizie costanti e di lunga durata diventano sempre 
piu rare e preziose auguriamoci che la nostra possa prosperare a lungo. 

Emilio Segre 



To Edoardo Amaldi, on His Sixtieth Birthday 



Rather than contributing a more or less interesting page of physics, I 
prefer to offer to this book a few personal recollections of a friendship which 
has lasted for over forty years. 

Its antecedents extend for two or three generations, originating in the 
small town of Bozzolo, near Mantua. My father used to tell me of his child- 
hood, about a century ago. I thus learned that when he had done some 
childish mischief, his parents told him that they would report it to Judge 
Amaldi, Edoardo's grandfather; this threat was supposed to make him more 
careful in the future. Our uncles were schoolmates in the distinguished 
Collegio Ghislieri at the University of Pavia; and even before meeting you, 
I knew your uncles and cousins. Furthermore, as an engineering student, 
I had enrolled in a course on the theory of a complex variable which your 
father gave at the University of Rome. It was such a fascinating course that 
I did not miss a lecture although it was not a required course and was given 
at an inconvenient time. 

We met in 1926 or shortly after and we went mountain climbing together. 
I am confident you will have by now digested but not forgotten the can of 
antipasti on the Corno Piccolo, in the Gran Sasso d'ltalia group. At that 
time — even as now — you were rosy complexioned and athletic. When we 
took some University courses together, you were entrusted, as you may 
remember, not only with the task of taking the superb notes with which we 
prepared for the examinations, but also with another delicate task; and it 
was your faithful help which allowed me to hear from Volterra himself the 
double delta squared and the theory of distortions. At that time we prepared 
for the examination at my parents' house in Tivoli. It was a simple job com- 
pared to what, I am told, you require from your present students. 

Study under Fermi from 1927 on was an unforgettable experience for 
all those who has the good luck of participating in it. For us it even has 
the added poignancy of covering quantum mechanics in its early formative 
phase. Without doubt this was the highlight of our cultural formation as 
physicists. In those halcyon years in Via Panisperna we also had visits from 
our foreign friends — first of all the unforgettable Placzek — from whom we 



XXII To Edoardo Amaldi, on his sixtieth birthday 

learned much and not only in physics, and furthermore Majorana, who once 
in a while stunned us with his groups. We turned to your father for help, 
because my experience as a student had convinced me that if there were any 
hope of mastering group theory, it was through his teaching. Contrary to 
our rather utilitarian approach, typical of young experimental physicists, 
he told us that we should really study also Lie groups. He was a prophet, 
and would rejoice seeing physicists using SU Z and beginning to appreciate 
what was one of his chief interests. 

Of our early experimental work in Rome I want to remind you only of 
the « swollen atoms », of the mercury shower we took and of our hydrogen 
arc for the ultraviolet and its explosion. Your precious ammonia Raman 
spectrum is of the same period. However, luckily not all experiments had 
such dramatic adventures. 

The neutron work with Fermi is by now historical. It gave us also an 
opportunity to spend some weeks at Cambridge and to meet Rutherford 
just when Ugo Jr. was about to be born. A little later, when I was already 
in Palermo, we spent a summer together at Alba di Canazei in the Dolomites, 
carrying our little children in baskets on the alpine meadows. 

Later we went through the tragic years of war and persecution. You 
were at my house in Berkeley when the war broke out, and I vividly remember 
the anxious days of the fatal September 1939 and you decision to return 
to Italy. 

Communications were interrupted for the duration of the war. The first 
news from Italy reached me in Los Alamos in 1944, where I was told that 
an American mission had been instructed to look for you on entering Rome. 

We saw each other again in 1947, the first time I returned to Rome after 
the war. War scars were visible everywhere, and the conversation centered 
on war events. I thus learned of your adventures in Africa, of your small 
rubber stamp factory to produce « unofficial » documents under the German 
occupation of Rome, and of sadder events. 

Later on, when the CERN project was in its formative stages, we discussed 
it in detail once more under the olive trees of Tivoli. Your enthusiasm for 
it, your arguments for the soundness of its approach to the problem of « big 
physics » and for its possible wider beneficial implications, carried conviction. 
Your visits to Berkeley interspersed with mine to Rome, followed. I think 
it is now your turn to come to Berkeley again. The last time you visited me 
at leisure was during the antiproton work, which gave us once more the 
occasion to write a paper together. 

At a time when long and constant friendships are becoming rare and 
precious, let us hope that ours may thrive yet for a long time. 

Emilio Segre 



SCIENTIFIC CONTRIBUTIONS 



Recent Developments in Particle Physics (*). 

L. W. Alvarez 
The Lawrence Radiation Laboratory - Berkeley, Cat. 



When I received my B. S. degree in 1932, only two of the fundamental 
particles of physics were known. Every bit of matter in the universe was 
thought to consist solely of protons and electrons. But in that same year, the 
number of particles was suddenly doubled. In two beautiful experiments, 
Chadwick [1] showed that the neutron existed, and Anderson [2] photogra- 
phed the first unmistakable positron track. In the years since 1932, the list 
of known particles has increased rapidly, but not steadily. The growth has 
instead been concentrated into a series of spurts of activity. 

Following the traditions of this occasion, my task this afternoon is to 
describe the latest of these periods of discovery, and to tell you of the devel- 
opment of the tools and techniques that made it possible. Most of us who 
become experimental physicists do so for two reasons; we love the tools of 
physics because to us they have intrinsic beauty, and we dream of finding 
new secrets of nature as important and as exciting as those uncovered by 
our scientific heroes. But we walk a narrow path with pitfalls on either side. 
If we spend all our time developing equipment, we risk the appellation of 
« plumber », and if we merely use the tools developed by others, we risk the 
censure of our peers for being parasitic. For these reasons, my colleagues 
and I are grateful to the Royal Swedish Academy of Science for citing both 
aspects of our work at the Lawrence Radiation Laboratory at the University 
of California — the observations of a new group of particles and the creation 
of the means for making those observations. 

As a personal opinion, I would suggest that modern particle physics started 
in the last days of World War II, when a group of young Italians, Conversi, 
Pancini, and Piccioni, who were hiding from the German occupying forces, 
initiated a remarkable experiment. In 1946, they showed [3] that the 
« mesotron », which had been discovered in 1937 by Neddermeyer and Ander- 
son [4] and by Street and Stevenson [5], was not the particle predicted by 



(*) Nobel Lecture, December 11, 1968. Copyright © The Nobel Foundation 1969. 



2 L. W. Alvarez 

Yukawa [6] as the mediator of nuclear forces, but was instead almost com- 
pletely unreactive in a nuclear sense. Most nuclear physicists had spent the 
war years in military-related activities, secure in the belief that the Yukawa 
mesons was available for study as soon as hostilities ceased. But they were 
wrong. 

The physics community had to endure less than a year of this nightmarish 
state; Powell and his collaborators [7] discovered in 1947 a singly charged 
particle (now known as the pion) that fulfilled the Yukawa prediction, and 
that decayed into the « mesotron », now known as the muon. Sanity was 
restored to particle physics, and the pion was found to be copiously produced 
in Ernest Lawrence's 184-inch cyclotron, by Gardner and Lattes [8] in 1948. 
The cosmic ray studies of Powell's group were made possible by the elegant 
nuclear emulsion technique they developed in collaboration with the Ilford 
laboratories under the direction of C. Waller. 

In 1950, the pion family was filled out with its neutral component by 
three independent experiments. In Berkeley, at the 184-inch cyclotron, Moyer, 
York, et al. [9] measured a Doppler-shifted y-ray spectrum that could only 
be explained as arising from the decay of a neutral pion, and Steinberger, 
Panofsky and Steller [10] made the case for this particle even more convincing 
by a beatiful experiment using McMillan's new 300 MeV synchrotron. And 
independently at Bristol, Ekspong, Hopper, and King [11] observed the two- 
y-ray decay of the n° in nuclear emulsion, and showed that its lifetime was 
less than 5-10~ 14 s. 

In 1952 Anderson, Fermi, and their collaborators [12] at Chicago started 
their classic experiments on the pion-nucleon interaction at what we would 
now call low energy. They used the external pion beams from the Chicago 
synchrocyclotron as a source of particles, and discovered what was for a long 
time called the pion-nucleon resonance. The isotopic spin formalism, which 
had been discussed for years by theorists since its enunciation in 1936 by 
Cassen and Condon [13], suddenly struck a responsive chord in the experi- 
mental physics community. They were impressed by the way Brueckner [14] 
showed that « /-spin » invariance could explain certain ratios of reaction 
cross-sections, if the resonance, which had been predicted many years earlier 
by Pauli and Dancoff [15] were in the f isotopic spin state, and had an angular 
momentum of f . 

By any test we can now apply, the « 3,3 resonance » of Anderson, Fermi 
et al. was the first of the « new particles » to be discovered. But since the rules 
for determining what constitutes a discovery in physics have never been 
codified — as they have been in patent law — it is probably fair to say that 
it was not customary, in the days when the properties of the 3,3 resonance 
were of paramount importance to the high energy physics community, to 



Recent developments in particle physics 3 

regard that resonance as a « particle ». Neutron spectroscopists study hun- 
dreds of resonances in neutron-nucleus system which they do not regard 
as separate entities, even though their lives are billions of times as long. 
I don't believe that an early and general recognition that the 3,3 resonance 
should be listed in the « table of particles » would in any way have speeded 
up the development of high energy physics. 

Although the study of the production and the interaction of pions had 
passed in a decisive way from the cosmic ray groups to the accelerator labora- 
tories in the late 1940's, the cosmic-ray-oriented physicists soon found two 
new families of « strange particles » — the K mesons and the hyperons. The 
existence of the strange particles has had an enormous inpact on the work 
done by our group at Berkeley. It is ironic that the parameters of the Beva- 
tron were fixed and the decision to build that accelerator had been made 
before a single physicist in Berkeley really believed in the existence of strange 
particles. But as we look back on the evidence, it is obvious that the observa- 
tions were well made, and the conclusions were properly drawn. Even if we 
had accepted the existence — an more pertinently the importance — of these 
particles, we would not have known what energy the Bevatron needed to 
produce strange particles; the associated production mechanism of Pais [16] 
and its experimental proof by Fowler, Shutt et al. [17] were still in the future. 
So the fact that, with a few notable exceptions, the Bevatron has made its 
greatest contributions to physics in the field of strange particles must be 
attributed to a very fortunate set of accidents. 

The Bevatron's proton energy of 6.3 GeV was chosen so that it would 
be able to produce antiprotons, if such particles could be produced. Since, 
in the interest of keeping the « list of particles » tractable, we no longer count 
antiparticles nor individual members of /-spin multiplets, it is becoming 
fashionable to regard the discovery of the antiproton as an « obvious exercise 
for the student ». (If we were to apply the « new rules » to the classical work 
of Chadwick and Anderson, we would conclude that they hadn't done anything 
either — the neutron is simply another /-spin state of the proton, and Ander- 
son's positron is simply the obvious antielectron !) In support of the non- 
obvious nature of the Segre group's discovery of the antiproton [18] I need 
only recall that one of the most distinguished high energy physicists I know, 
who didn't believe that antiprotons could be produced, was obliged to settle 
a 500-dollar bet with a colleague who held the now universally accepted 
belief that all particles can exist in an antistate. 

I have just discussed in a very brief way the discovery of some particles 
that have been of importance in our bubble chamber studies, and I will 
continue the discussion throughout my lecture. This account should not be 
taken to be authoritative — there is no authority in this area — but simply 



4 L. W. Alvarez 

as a narrative to indicate the impact that certain experimental work had 
on my own thinking and on that of my colleagues. 

I will now return to the story of the very important strange particles. 
In contrast to the discovery of the pion, which was accepted immediately 
by almost everyone — one apparent exception will be related later in this talk — 
the discovery and the eventual acceptance of the existence of the strange 
particles stretched out over a period of a few years. Heavy, unstable particles 
were first seen in 1947, by Rochester and Butler [19], who photographed 
and properly interpreted the first two « V particles » in a cosmic-ray-triggered 
cloud chamber. One of the V's was charged, and was probably a K meson. 
The other was neutral, and was probably a K°. For having made these 
observations, Rochester and Butler are generally credited with the discovery 
of strange particles. There was a disturbing period of two years in which 
Rochester and Butler operated their chamber and no more V particles were 
found. But in 1950 Anderson, Leighton et al. [20] took a cloud chamber 
to a mountain top and showed that it was possible to observe approximately 
one V particle per day under such conditions. They reported, « To interpret 
these photographs, one must come to the same remarkable conclusion as that 
drawn by Rochester and Butler on the basis of these two photographs, viz., 
that these two types of events represent, respectively, the spontaneous decay 
of neutral and charged unstable particles of a new type ». 

Butler and his collaborators then took their chamber to the Pic-du-Midi 
and confirmed the high event rate seen by the CalTech group on White 
Mountain. In 1952 they reported the first cascade decay [21] — now known 
as the E~ hyperon. 

While the cloud chamber physicists were slowly making progress in under- 
standing the strange particles, a parallel effort was under way in the nuclear 
emulsion-oriented laboratories. Although the first K meson was undoubtedly 
observed in Leprince-Ringuet's cloud chamber [22] in 1944, Bethe [23] cast 
sufficient doubt on its authenticity that it had no influence on the physics 
community and on the work that followed. The first overpowering evidence 
for a K meson appeared in nuclear emulsion, in an experiment by Brown 
and most of the Bristol group [24], in 1949. This so-called t + meson decayed 
at rest into three coplanar pions. The measured ranges of the three pions 
gave a very accurate mass value for the t meson of 493.6 MeV. Again there 
was a disturbing period of more than a year and a half before another t meson 
showed up. 

In 1951, the year after the t meson and the V particles were finally seen 
again, O'Ceallaigh [25] observed the first of his kappa mesons in nuclear 
emulsion. Each such event involved the decay at rest of a heavy meson into 
a muon with a different energy. We now know these particles as K + mesons 



Recent developments in particle physics 5 

decaying into fx + +7c°+v, so the explanation of the broad muon energy 
spectrum is now obvious. But it took some time to understand this in the 
early 1950's, when these particles appeared one by one in different laboratories. 
In 1953, Menon and O'Ceallaigh [26] found the first K^ or 8 meson, with 
a decay into ~ + +tt . The identification of the 6 and t mesons as different 
decay modes of the same K mesons is one of the great stories of particle 
physics, and it will be mentioned later in this lecture. 

The identification of the neutral A emerged from the combined efforts 
of the cosmic ray cloud chamber groups, so I will not attempt to assign 
credit for its discovery. But it does seem clear that Thompson et ah [27] 
were the first to establish the decay scheme of what we now know as the K? 
meson: Ki^tc++7tT. The first example of a charged 21 hyperon was seen 
in emulsion by the Genoa and Milan group [28], in 1953. And after that, 
the study of strange particles passed, to a large extent, from the cosmic ray 
groups to the accelerator laboratories. 

So by the time the Bevatron first operated, in 1954, a number of different 
strange particles had been identified ; several charged particles and a neutral 
one all with masses in the neighborhood of 500 MeV, and three kinds of 
particles heavier than the proton. In order of increasing mass, these were 
the neutral A, the two charged 2's (plus and minus), and the negative cascade 
(E~), which decayed into a A and a negative pion. 

The strange particles all had lifetimes shorter than any known particles 
except the neutral pion. The hyperons all had lifetimes of approximately 
10~ 10 s, or less than 1 % of the charged pion lifetime. When I say that 
they were called strange particles because their observed lifetimes presented 
such a puzzle for theoretical physicists to explain, I can imagine the lay 
members in this audience saying to themselves, « Yes, I cannot see how 
anything could come apart so fast. » But the strangeness of the strange 
particles is not that they decay so rapidly, but that they last almost a million 
million times longer than they should — physicists could not explain why 
they did not come apart in about 10~ 21 s. 

I will not go into the details of the dilemma, but we can note that a similar 
problem faced to physics community when the muon was found to be so 
inert, nuclearly. The suggestion by Marshak and Bethe [29] that it was the 
daughter of a strongly interacting particle was published almost simultane- 
ously with the independent experimental demonstration by Powell et ah 
mentioned earlier. Although invoking a similar mechanism to bring order 
into the strange-particle arena was tempting, Pais [16] made his suggestion 
that strange particles were produced « strongly » in pairs, but decayed 
« weakly » when separated from each other. 

Gell-Mann [30] (and independently Nishijima [31] then made the first 



6 L. W. Alvarez 

of this several major contributions to particle physics by correctly guessing 
the rules that govern the production and decay of all the strange particles. 
I use the word « guessing » with the same sense of awe I feel when I say that 
Champollion guessed the meanings of the hieroglyphs on the Rosetta Stone. 
Gell-Mann had first to assume that the K meson was not an /-spin triplet, 
as it certainly appeared to be, but an /-spin doublet plus is antiparticles, and 
he had further to assume the existence of the neutral 2 and fo the neutral E. 
And finally, when he assigned appropriate values of his new quantum 
number, strangeness, to each family, his rules explained the one observed 
production reaction and predicted a score of others. And of course it 
explained all the known decays, and predicted another. My research group 
eventually confirmed all of Gell-Mann's and Nishijima's early predictions, 
many of them for the first time, and we continue to be impressed by their 
simple elegance. 

This was the state of the art in particle physics in 1954, when William 
Brobeck turned his brainchild, the Bevatron, over to his Radiation Laboratory 
associates to use as a source of high energy protons. I has been using the 
Berkeley proton linear accelerator in some studies of short-lived radioactive 
species, and I was pleased at the chance to switch to a field that appeared to 
be more interesting. My first Bevatron experiment was done in collaboration 
with Sula Goldhaber [32] ; it gave the first real measurement of the t meson 
lifetime. My next experiment was done with three talented young post- 
doctoral fellows, Frank S. Crawford jr., Myron L. Good and M. Lynn 
Stevenson. An early puzzle in K-meson physics was that two of the particles 
(the 6 and t) had similar, but poorly determined lifetimes and masses. That 
story has been told in this auditorium by Lee [33] and Yang [34] so I will 
not repeat it now. But I do like to think that our demonstration [35], simul- 
taneously with and independently from one by Fitch and Motley [36], that 
the two lifetimes were not measurably different, plus similar small limits on 
possible mass differences set by von Friesen et al. [37] and by Birge et al. [38], 
nudged Lee and Yang a bit toward their revolutionary conclusion. 

Our experiences with what was then a very complicated array of scintilla- 
tion counters led me and my colleagues to despair of making meaningful 
measurements of what we perceived to be the basic reactions of strange particle 
physics : 

7i-+p ->A + K° 

I I 

the production reaction is indicated by the horizontal arrows, the subsequent 
decays by the vertical arrows. Figure 1 shows a typical example of this reac- 



Recent' developments in particle physics 




Fig. J. 



+-»-* K"+A. 



tion, as we saw it later in the 10 in. bubble chamber. We concluded, correctly 
I believe, that none of the then known techniques was well suited to study 
this reaction. Counters appeared hopelessly inadequate to the task, and the 
spark chamber had not yet been invented. The Brookhaven diffusion cloud 
chamber group [17] had photographed only a few events like shown in 
Fig. I, in a period of two years. It seemed to us that a track-recording 
technique was called for, but each of the three known track devices had 
drawbacks that ruled it out as a serious contender for the role we envisaged. 
Nuclear emulsion, which had been so spectacularly successful in the hands 
of Powell's group, depended on the contiguous nature of the successive tracks 
at a production or decay vertex. The presence of neutral and therefore non- 
ionizing particles between related charged particles, plus lack of even a rudi- 
mentary time resolution, made nuclear emulsion techniques virtually unusable 
in this new field. The two known types of cloud chambers appeared to have 
equally insurmountable difficulties. The older Wilson expansion chamber 
had two difficulties that rendered it unsuitable for the job: if used at atmos- 
pheric pressure, its cycling period was measured in minutes, and if one 
increased its pressure to compensate for the long mean free path of nuclear 
interactions, its cycling period increased at least as fast as the pressure was 
increased. Therefore the number of observed reactions per day started at 



8 L. W. Alvarez 

an almost impossibly low value, and dropped as « corrective action » was 
taken. The diffusion cloud chamber was plagued by « background problems », 
and had an additional disadvantage — its sensitive volume was confined in 
the vertical direction to a height of only a few centimeters. What we conclude 
from all this was simply that particle physicists needed a track-recording device 
with solid or liquid density (to increase the rate of production of nuclear 
events by a factor of 100), with uniform sensitivity (to avoid the problems 
of the sensitive layer in the diffusion chamber), and with fast cycling time 
(to avoid the Wilson chamber problems). And of course, any cycling detector 
would permit the association of charged tracks joined by neutral tracks, 
which was denied to the user of nuclear emulsion. 

In late April of 1953 I paid my annual visit to Washington, to attend the 
meeting of the American Physical Society. At lunch of the first day, I found 
myself seated at a large table in the garden of the Shoreham Hotel. All the 
seats but one were occupied by old friends from World War II days, and we 
reminisced about our experiences at the MIT radar laboratory and at Los 
Alamos. A young chap who had not experienced those exciting days was 
seated at my left, and we were soon talking of our interests in physics. He 
expressed concern that no one would hear his 10 min contributed paper, 
because it was scheduled as the final paper of the Saturday afternoon ses- 
sion, and therefore the last talk to be presented at the meeting. In those 
days of slow airplanes, there were even fewer people in the audience for the 
last paper of the meeting than there are now — if that is possible. I admitted 
that I would not be there, and asked him to tell me what he would be reporting. 
And that is how I heard first hand from Donald Glaser how he had invented 
the bubble chamber, and to what state he had brought its development. 
And of course he has since described those achievements from this plat- 
form [39]. He showed me photographs of bubble tracks in a small glass 
bulb, about 1 cm in diameter and 2 cm long, filled with diethyl ether. He 
stressed the need for absolute cleanliness of the glass bulb, and said that 
he could maintain the ether in a superheated state for an average of many 
seconds before spontaneous boiling took place. I was greatly impressed by 
his work, and it immediately occurred to me that this could be the « big 
idea » I felt was needed in particle physics. 

That night in my hotel room I discussed what I had learned with my col- 
league from Berkeley, Frank Crawford. I told Frank that I hoped we could 
get started on the development of a liquid hydrogen chamber, much larger 
than anything Don Glaser was thinking about, as soon as I returned to Berke- 
ley. He volunteered to stop off in Michigan on the way back to Berkeley, 
which he did, and learned everything he could about Glaser's technique. 

I returned to Berkeley on Sundary, May 1, and on the next day Lynn 



Recent developments in particle physics 9 

Stevenson started to keep a new notebook on bubble chambers. The other day, 
when he saw me writing this talk, he showed me that old notebook with its 
first entry dated May 2, 1953, with Van der Waal's equation on the first 
page, and the isotherms hydrogen traced by hand onto the second page. 
Frank Crawford came home a few days later, and he and Lynn moved into 
the « student shop » in the synchrotron building, to build their first bubble 
chamber. They were fortunate in enlisting the help of John Wood who was 
an accelerator technician at the synchrotron. The three of them put their 
first efforts into a duplication of Glaser's work with hydrocarbons. When they 
has demonstrated radiation sensitivity in ether, they built a glass chamber in 
a Dewar flask to try first with liquid nitrogen and then with liquid hydrogen. 

I remember that on several occasions I telephoned to the late Earl Long 
at the University of Chicago, for advice on cryogenic problems. Dr. Long 
gave active support to the liquid hydrogen bubble chamber that was being 
built at that time by Roger Hildebrand and Darragh Nagle at the Fermi 
Institute in Chicago. In August of 1953 Hildebrand and Nagle [40] showed 
that superheated hydrogen boiled faster in the presence of a gamma-ray source 
than it did when the source was removed. This is a necessary (though not 
sufficient) condition for successful operation of a liquid hydrogen bubble 
chamber, and the Chicago work was therefore an important step in the devel- 
opment of such chambers. The important unanswered question concerned 
the bubble density— was it sufficient to see tracks of « minimum ionizing » 
particles, or did liquid hydrogen — as my colleagues had just shown that 
liquid nitrogen did — produce bubbles but no visible tracks? 

John Wood saw the first tracks in a 1.5 in.-diameter liquid hydrogen bub- 
ble chamber in February of 1954 [41]. The Chicago group could certainly 
have done so earlier, by rebuilding their apparatus, but they switched their 
efforts to hydrocarbon chambers, and were rewarded by being the first physi- 
cists to publish experimental results obtained by bubble chamber techniques. 
Figure 2 is a photograph of Wood's first tracks. 

At the Lawrence Radiation Laboratory, we have long had a tradition of 
close cooperation between physicists and technicians. The resulting at- 
mosphere, which contributed so markedly to the rapid development of the 
liquid hydrogen bubble chamber, led to an unusual phenomenon: none of the 
scientific papers on the development of bubble chamber techniques in my 
research group were signed by experimenters who were trained as physicists 
or who had had previous cryogenic experience. The papers all had authors 
who were listed on the Laboratory records as technicians, but of course the 
physicists concerned knew what was going on, and offered many suggestions. 
Nonetheless, our technical associates carried the main responsibility, and 
published their findings in the scientific literature. I believe this is a healthy 



10 L. W. Alvarez 




Fig. 2. First tracks in hydrogen. 

change from practices thai were common a generation ago; we all remember 
papers signed by a single physicist that ended with a paragraph saying, « 1 wish 

to thank Mr. , who built the apparatus and took much of the 

data ». 

And speaking of acknowledgments, John Wood's first publication, in ad- 
dition to thanking Crawford, Stevenson, and me for our advice and help, said, 
«1 am indebted to A. J. Schwcmin for help with the electronic circuits ». 
« Pete » Schwemin, the most versatile technician I have ever known, became 
so excited by his initial contact with John Wood's 1.5 in. -diameter all-glass 
chamber tluit he immediately started the construction of" the first metal 
bubble chamber with glass windows. All earlier chambers had been made 
completely of smooth glass, without joints, to prevent accidental boiling at 
sharp points; such boiling of course destroyed the superheat and made the 
chamber insensitive to radiation. Both Glaser and Hildcbrami stressed the 
long times their liquids could be held in the superheated condition; Hilde- 
brand and Nagle averaged 22 s and observed one superheat period of 70 s. 
John Wood reported [41], « Wc were discouraged by our inability to attain 
the long times of superheat, until the track photographs showed that it was 
not important in the successful operation of a large bubble chamber ». 1 have 
always felt that second to Glaser's discovery of tracks this was the key 



Recent developments in particle physics 1 1 

observation in the whole development of bubble chamber technique. As long 
as one « expanded the chamber » rapidly, bubbles forming on the wall didn't 
destroy the superheated condition of the main volume of the liquid, and it 
remained sensitive as a track-recording medium. 

Pete Schwemin, with the help of Douglas Parmentier [42], built the 2.5 in.- 
diameter hydrogen chamber in record time, as the world's first « dirty cham- 
ber ». I have never liked that expression, but it was used for a while to 
distinguish chambers with windows gasketed to metal bodies from all-glass 
chambers. Because of it « dirtiness », the 2.5 in. chamber boiled at its walls, 
but still showed good tracks throughout its volume. Now that « clean » 
chambers are of historical interest only, we can be pleased that the modern 
chambers need no longer be stigmatized by the adjective « dirty ». 

Lynn Stevenson's notebook shows a diagram of John Wood's chamber 
dated January 25, 1954, with Polaroid pictures of tracks in hydrogen. A 
month later he recorded details of Schwemin's 2.5 in. chamber, and drew 
a complete diagram dated March 5. (That was the day after the Physical 
Review received Wood's letter announcing the first observation of tracks.) 
On April 29, Schwemin and Parmentier photographed their first tracks ; these 
are shown in Fig. 3. (Things were happening so fast at this time that the 
2.5 in. system was never photographed as a whole before it ended up on the 
scrap pile.) 

In August, Schwemin and Parmentier separately built two different 4 in.- 
diameter chambers. Both were originally expanded by internal bellows, and 
Parmentier's 4 in. chamber gave tracks on October 6. Schwemin's chamber 
produced tracks three weeks later, and survived as the 4 in. chamber. (See 
Fig. 4.) The bellows systems in both chambers failed, but it turned out to be 
easier to convert Schwemin's chamber to the vapor expansion system that 
was used in all our subsequent chambers until 1962. (In that year, the 25 in. 
chamber introduced the « Q bellow » that is now standard for large chambers.) 

Figure 5 shows all our chambers displayed together a few weeks ago, at 
the request of Swedish Television. As you can see, we all look pretty pleased 
to see so many of our « old friends » side by side for the first time. 

Figure 6 shows an early picture of multiple meson production in the 4 in. 
chamber. This chamber was soon equipped with a pulsed magnetic field, and 
in that configuration it was the first bubble chamber of any kind to show 
magnetically curved tracks. It was then set aside by our group as we pushed 
on to larger chambers. But it ended its career as a useful research tool at 
the Berkeley electron synchrotron, after almost two million photographs of 
300 MeV bremsstrahlung passing through it had been taken and analyzed 
by Bob Kenney et al. [43]. 

In the year 1954, as I have just recounted, various members of my research 



12 



L, W, Akcrei 




I 




Recent deicfojments in par tick physics 13 




Fig, 4. - 4 in, chamber, D, Parmenlier (left), A, J, Sehwemrn (right), 



group had been responsible for the successful operation of four separate 
liquid hydrogen bubble chambers, increasing in diameter from 1-5 to 4 in. 
By the end of that eventful year, it was clear that it would take a more con- 
certed engineering-type approach to the problem if we were to progress to the 
larger chambers we felt were essential to the solution of high-energy physics 
problems. 1 therefore enlisted the assistance of three close associates, J. Donald 
Gow, Robert Walt and Richard Bluniberg. Don Gow and Bob Watt had 



14 



L. W. Alvarez 




Fig. 5. - Display of chambers, November 1968. Prom left to right, 1^, 4, 6, 
10, 15 and 72 in. chambers; Hernandez, Schwemin, Rinla, Watt,, Alvarez and 

Eckman, 



taken over full responsibility for the development and operation of the 
32 McV linear accelerator that had occupied ail my attention from its inception 
late in 1945 until it first operated in late 1947. Neither of them had any 
experience with cryogenic techniques, but they learned rapidly, and were soon 
leaders in the new technology of hydrogen bubble chambers. Dick Blum berg 
had been trained as a mechanical engineer, and he had designed the equip- 
ment used by Crawford, Stevenson and me in our experiments, then in pro- 
gress, on the Compton scattering of y-rays by protons [44]. 

Wilson Powell had built two large magnets to accommodate his Wilson 
Cloud Chambers, pictures from which adorned the walls of every cyclotron 
laboratory in the world. He very generously plaeed one of these magnets at 
our disposal, and Dick. Blum berg immediately started the mechanical design 
of the 10 in. chamber— the largest size we felt could be accommodated in 
the well of Powell's magnet, Blumberg's drafting table was in the middle of 
the single room that contained the desks of all the members of my research 
group. Not many engineers will tolerate such working conditions, but Blum- 
berg was able to do so and he produced a design that was quickly built in 
the main machine shop. All earlier chambers had been built by the exper- 
imenters themselves. The design of the 10 in. chamber turned out to be a 
much larger job than we had foreseen. By the time it was completed, eleven 
members of the Laboratory's Mechanical Engineering Department had worked 
on it, including Rod Byrns and John Mark. The electrical engineering aspects 
of all our large chambers were formidable, and we are indebted to Jim Shand 
for his leadership in this work for many years. 



Recent developments in particle physics 



15 




Fig, 6, - Multiple meson production in 4 in. chamber. 



Great difficulty was experienced with the first operation of the 10 in. 
chamber; too much hydrogen was vaporized at each « expansion ». Pete 
Schwemin quickly diagnosed the trouble and built a fast-acting valve that 
permitted the chamber to be pulsed every 6 s, to match (lie Bevatron's 
cycling time. 

It would be appropriate to interrupt this description of the bubble chamber 
development program to describe the important observations made possible 
by the operation of the 10 in. chamber early in 1956, but instead, I will 
preserve the continuity by describing the further development of the hardware. 
In December of 1954, shortly after the 4 in. chamber had been operated in 
the cyclotron building for the first time, it became evident to me that the 
10 in. chamber we bad just started to design would not be nearly large 
enough to tell us what we wanted to know about the strange particles. The 
tracks of these objects had been photographed at Brookhaven [17], and we 
knew they were produced copiously by the Bevatron, 



16 L. W. Alvarez 

The size of the « big chamber » was set by several different criteria, and 
fortunately all of them could be satisfied by one design. (Too often, a designer 
of new equipment finds that one essential criterion can be met only if the 
object is very large, while an equally important criterion demands that it 
be very small.) All « dirty chambers » so far built throughout the world had 
been cylindrical in shape, and were characterized by their diameter measure- 
ment. By studying the relativistic kinematics of strange particles produced by 
Bevatron beams, and more particularly by studying the decay of these par- 
ticles, I convinced myself that the big chamber should be rectangular, with a 
length of at least 30 in. This length was next increased to 50 in. in order that 
there would be adequate amounts of hydrogen upstream from the required 
decay region, in which production reactions could take place. Later the length 
was charged to 72 in. , when it was realized that the depth of the chamber 
could properly be less than its width and that the change could be made 
without altering the volume. The production region corresponded to about 
10% of a typical pion-proton mean free path, and the size of the decay region 
was set by the relativistic time-dilated decay lengths of the strange particles, 
plus the requirement that there be a sufficient track length available in which 
to measure magnetic curvature in a « practical magnetic field » of 15000 G. 
In summary, then, the width and depth of the chamber came rather simply 
from an examination of the shape of the ellipses that characterize relativistic 
transformations at Bevatron energies, plus the fact that the magnetic field 
spreads the particles across the width but not along the 'depth of the chamber. 

The result of this straightforward analysis was a rather frightening set of 
numbers: The chamber length was 72 in.; its width was 20 in., and its 
depth was 15 in. It had to be pervaded by a magnetic field of 15000 G, 
so its magnet would weigh at least 100 tons and would require 2 or 3 MW 
to energize it. It would require a window 75 in. long by 23 in. wide and 5 in. 
thick to withstand the (deuterium) operating pressure of 8 atm, exerting a 
force of 100 tons on the glass. No one had any experience with such large 
volumes of liquid hydrogen; the hydrogen-oxygen rocket eingines that now 
power the upper stages of the Saturn boosters were still gleams in the eyes 
of their designers — these were pre-Sputnik days. The safety aspects of the 
big chamber were particularly worrisome. Low temperature laboratories had 
a reputation for being dangerous places in which to work, and they did not 
deal with such large quantities of liquid hydrogen, and what supplies they 
did use were kept at atmospheric pressure. 

For some time, the glass window problem seemed insurmountable — no 
one had ever cast and polished such a large piece of optical glass. Fortunately 
for the eventual success of the project, I was able to persuade myself that 
the chamber body could be constructed of a transparent plastic cylinder with 



Recent developments in particle physics 17 

metallic end plates. This notion was later demolished by my engineering 
colleagues, but it played an important role in keeping the project alive in 
my own mind until I was convinced that the glass window could be built. 
As an indication of the cryogenic « state of the art » at the time we worried 
about the big window, I can recall the following anecdote. One day, while 
looking through a list of titles of talks at a recent cryogenic conference, I 
spotted one that read, « Large glass window for viewing liquid hydrogen ». 
Eagerly I turned to the paper — but it described a metallic Dewar vessel 
equipped with a glass window 1 in. in diameter! 

Don Gow was now devoting all his to hydrogen bubble chambers, and 
in January of 1955 we interested Paul Hernandez in taking a good hard 
engineering look at the problems involved in building and housing the 72 in. 
bubble chamber. We were also extremely fortunate in being agle to interest the 
cryogenic engineers at the Boulder, Colorado, branch of the National Bureau 
of Standards in the project. Dudley Chelton, Bascomb Birmingham and Doug 
Mann spent a great deal of time with us, first educating us in large-scale 
liquid hydrogen techniques, and later cooperating with us in the design and 
initial operation of the big chamber. 

In April of 1955, after several months of discussion of the large chamber, 
I wrote a document entitled The Bubble Chamber Program at UCRL. This 
paper showed in some detail why it was important to build the large chamber, 
and outlined a whole new way of doing high-energy physics with such a device 
It stressed the need for semiautomatic measuring devices (which had not 
previously been proposed), and described how electronic computers would 
reconstruct tracks in space, compute momenta, and solve problems in rela- 
tivistic mechanics. All these techniques are now part of the « standard bubble 
chamber method », but in April of 1955 no one had yet applied them. Of all 
the papers I have written in my life, none gives me so much satisfaction on 
rereading as does this unpublished prospectus. 

After Paul Hernandez and Don Gow has estimated that the big chamber, 
including its building and power supplies, would cost about 2.5 million dollars, 
it was clear that a special AEC appropriation was required; we could no 
longer build our chambers out of ordinary laboratory operating money. In 
fact, the document I have just described was written as a sort of proposal 
to the AEC for financial support — but without mentioning money! I asked 
Ernest Lawrence if he would help me in requesting extra funds from the 
AEC. He read the document, and agreed with the points I had made. He 
then asked me to remind him of the size of the world's largest hydrogen 
chamber. When I replied that it was 4 in. in diameter, he said the though I 
was making too large an extrapolation in one step, to 72 in. I told him that 
the 10 in. chamber was on the drawing board, and if we could make it work, 



18 L. W. Alvarez 

the operation of the 72 in. chamber was assured. (And if we could not make 
it work, we could refund most of the 2.5 million.) This was not obvious 
until I explained the hydraulic aspects of the expansion system of the 72 in. 
chamber; it was arranged so that the 20 in. wide, 72 in. long chamber could 
be considered to be a large collection of essentially independently expanded 
10 in. square chambers. He was not convinced of the wisdom of the pro- 
gram, but in a characteristic gesture, he said, « I don't believe in your big 
chamber, but I do believe in you, and I'll help you to obtain the money ». 
I therefore accompanied him on his next trip to Washington, and we talked 
in one day to three of the five Commissioners: Lewis Strauss, Willard Libby 
(who later spoke from this podium), and the late John von Neumann, the 
greatest mathematical physicist then living. That evening, at a cocktail party 
at Johnny von Neumann's home, I was told that the Commission had voted 
that afternoon to give the laboratory the 2.5 million dollars we had requested. 
All we had to do now was build the thing and make it work! 

Design work had of course been under way for some time, but it was 
now rapidly accelerated. Don Gow assumed a new role that is not common in 
physics laboratories, but is well known in military organizations; he became 
my « chief of staff ». In this position, he coordinated the efforts of the phys- 
icists and engineers; he had full responsibility for the careful spending of our 
precious 2.5 million dollars, and he undertook to become an expert second 
to none in all the technical phases of the operation, from low temperature 
thermodynamics to safety engineering. His success in this difficult task can 
be recognized most easily in the success of the whole program, culminating 
in the fact that I am speaking here this afternoon. I am sorry that Don Gow 
can not be here today; he died several years ago, but I am reminded of him 
every day — my three-year-old son is named Donald in his memory. 

The engineering team under Paul Hernandez's direction proceeded rapidly 
with the design, and in the process solved a number of difficult problems 
in ways that have become standard « in the industry ». A typical problem 
involved the very considerable differential expansion between the stainless 
steel chamber and the glass window. This could be lived with in the 10 in. 
chamber, but not in the 72 in. Jack Franck's « inflatable gasket » allowed 
the glass to be seated against the chamber body only after both had been 
cooled to liquid hydrogen temperature. 

Just before leaving for Stockholm, I attended a ceremony at which Paul 
Hernandez was presented with a trophy honoring him as a « Master Designer » 
for his achievements in the engineering of the 72 in. chamber. I had the 
pleasure of telling in more detail than I can today of his many contributions 
to the success of our program. One of his associates recalled a special service 
that he rendered not only to our group but to all those who followed us in 



Recent developments in particle physics 19 

building liquid hydrogen bubble chambers. Hernandez and his associates 
wrote a series of Engineering Notes, on matters of interest to designers of 
hydrogen bubble chambers, that soon filled a series of notebooks that spanned 
3 ft of shelf space. Copies of theses were sent to all interested parties on both 
sides of the Atlantic, and I am sure that they resulted in a cumulative savings 
to all bubble chamber builders of several million dollars ; had not all this infor- 
mation been readily available, the test programs and calculations of our 
engineering group would have required duplication at many laboratories, 
at a large expense of money and time. Our program moved so rapidly that 
there was never time to put the Engineering Notes into finished form for pub- 
lication in the regular literature. For this reason, one can now read review 
articles on bubble chamber technology, and be quite unaware of the part 
that our Laboratory played in its development. There are no references to 
papers by members of our group, since those papers were never written — the 
data that would have been in them had been made available to everyone who 
needed them at a much earlier date. 

And just to show that I was also deeply involved in the chamber design, 
I might recount how I purposely « designed myself into a corner » because I 
thought the result were important, and I thought I could invent a way out 
of a severe difficulty, if given the time. All previous chambers had had two 
windows, with « straight through » illumination. Such a configuration reduces 
the attainable magnetic field, because the existence of a rear pole piece would 
interfere with the light-projection system. I made the decision that the 72 in. 
chamber would have only a top window, thereby permitting the magnetic 
field to be increased by a lower pole piece and at the same time saving the 
cost of the extra glass window, and also providing added safety by eliminating 
the possibility that liquid hydrogen could spill through a broken lower window. 
The only difficulty was that for more than a year, as the design was firmed 
up and the parts were fabricated, none of us could invent a way both to 
illuminate and to photograph the bubbles through the same window. Duane 
Norgren, who has been responsible for the design of all our bubble chamber 
cameras, discussed the matter with me at least once a week in that critical 
year, and we tried dozens of schemes that did not quite do the job. But as a 
result of our many failures, we finally came to understand all the problems, 
and we eventually hit on the retrodirecting system known as coat hangers. 
This solution came none too soon ; if it had been delayed by a month or more, 
the initial operation of the 72 in. chamber would have been correspondingly 
delayed. We took many other calculated risks in designing the system; if 
we had postponed the fabrication of the major hardware until we had solved 
all the problems on paper, the project might still not be completed. Engineers 
are conservative people by nature; it is the ultimate disgrace to have a boiler 



20 L. W. Alvarez 

explode or a bridge collapse. We were therefore fortunate to have Paul 
Hernandez as our chief eingineer; he would seriously consider anything his 
physics colleagues might suggest, no matter how outlandish it might seem 
at first sight. He would firmly reject it if it could not be made safe, but before 
rejecting any idea for lack of safety he would use all the ingenuity he pos- 
sessed to make it safe. 

We felt that we needed to build a test chamber to gain experience with a 
single-window system, and to learn to operate with a hydrogen refrigerator; 
our earlier chambers had all used liquid hydrogen as a coolant. We therefore 
built and operated the 15 in. chamber in the Powell magnet, in place of the 
10 in. chamber that had served us so well. 

The 72 in. chamber operated for the first time on March 24, 1959, very 
nearly four years from the time it was first seriously proposed. Figure 7 shows 
it at about that time. The « start-up team » consisted of Don Gow, Paul 
Hernandez and Bob Watt, all of whom had played key roles in the initial 
operation of the 15 in. chamber. Bob Watt and Glenn Eckman have been 
responsible for the operation of all our chambers from the earliest days of 
the 10 in. chamber, and the success of the whole program has most often 
rested in their hands. They have maintained an absolutely safe operating 
record in the face of very severe hazard, and they have supplied their col- 
leagues in the physics community with approximately ten million high-quality 
stereo photographs. And most recently, they have shown that they can design 
chambers as well as they have operated them. The 72 in. chamber was 
recently enlarged to an 82 in. size, incorporating to a large extent the design 
concepts of Watt and Eckman. 

Although I have not done justice to the contributions of many close friends 
and associates who shared in our bubble chamber development program, I 
must now turn to another important phase of our activities — the data-analysis 
program. Soon after my 1955 prospectus was finished, Hugh Bradner under- 
took to implement the semiautomatic measuring machine proposal. He first 
made an exhaustive study of commercially available measuring machines, 
encoding techniques, etc., and then, with Jack Franck, designed the first 
« Franckenstein ». This rather revolutionary device has been widely copied, to 
such an extent that objects of its kind are now called « conventional » measur- 
ing machines (Fig. 8). Our first Franckenstein was operating reliably in 1957, 
and in the summer of 1958 a duplicate was installed in the U.S. exhibit at 
the « Atoms for Peace » exposition in Geneva. It excited a great deal of 
interest in the high-energy physics community, and a number of groups set 
out to make similar machines based on its design. Almost everyone thought 
at first that our provision for automatic track following was a needless waste 
of money, but over the years, that feature has also come to be « conventional ». 



Recent developments in particle physics 



21 



Jack Franck then went on to design the Mark II Franckenstein, to measure 
72 in. bubble chamber film. He had the first one ready to operate just in time 

to match the rapid turn-on of the big chamber, and he eventually built three 




Fig. 7. - 72 in. bubble chamber in its building. 



more of the Mark It's. Other members of our group then designed and 
perfected the faster and less expensive SMP system, which added significantly 
to our « measuring power ». The moving forces in this development were 
Pete Schwemin, Bob Hulsizer, Peter Davey, Ron Ross and Bill Humphrey [45] 



22 



L. W. Alvarez 



Our final and most rewarding effort to improve our measuring ability was 
fulfilled several years ago, when our first Spiral Reader became operational. 
This single machine has now measured more than one and a half million high 




Fig. 8, - « Franckcnstein ». 



energy interactions, and has, together with its almost identical twin, measured 
one and a quarter million events in the last year. The SAAB Company here 
in Sweden is now building and selling Spiral Readers to European laboratories. 
The Spiral Reader had a rather checkered career, and it was on several 
occasions believed by most workers in the field to have been abandoned by 
our group. The basic concept of the spiral scan was supplied by Bruce McCor- 
mick, In J 956. Our attempts to reduce his ideas to practice resulted in failure, 
and shortly after that, McCormick moved to Illinois, where he has since been 
engaged in computer development. As the cost of transistorized circuits 
dropped rapidly in the next years, we tried a second time to implement the 
Spiral Reader concept, using digital techniques to replace the analog devices 
of the earlier machine. The second device showed promise, but its « hard- 
wired logic » made it too inflexible, and the unreliability of its electronic 
components kept it in repair most of the time. The mechanical and optical 
components of the second Spiral Reader were excellent, and we hated to 
drop the whole project simply because the circuitry did not come up to the 



Recent developments in particle physics 



23 



same standard. In 1963, Jack Lloyd suggested that we use one of the new 
breed of small high-speed, inexpensive computers to supply the logic and the 
control circuits for the Spiral Reader. He then demonstrated great qualities 
of leadership by delivering to our research group a machine that has per- 
formed even better than he had promised it would. In addition to his develop- 
ment of the hardware, he initiated POOH, the Spiral Reader filtering program, 
which was brought to a high degree of perfection by Jim Burkhard. The 
smooth and rapid transition of the Spiral Reader from a developmental stage 
into a useful operational tool was largely the result of several years of hard 
work on the part of Gerry Lynch and Frank Solmitz. Figure 9, from a talk 
I gave two and a half years ago [46], shows how the measuring power of our 
group has increased over the years, with only a modest increase in personnel. 



,000,000 
800,000 - 

600,000 
400,000 - 



q. 200,000 - 
(/> 

c 

i2 io o,ooo 

80,000 
60,000 

40,000- 
20,000 - 
1 0,000 



200. 



- 100 




0) k. 

E°- 



Fig. 9. - Measuring rates. 



24 L. W. Alvarez 

According to a simple extrapolation of the exponential curve we had been 
on from 1957 through 1966, we would expect to be measuring 1.5 million 
events per year some time in 1969. But we have already reached that rate 
and we will soon be leveling off about there because we have stopped our 
development work in this area. 

The third key ingredient of our development program has been the 
continually increasing sophistication in our utilization of computers, as they 
have increased in computational speed and memory capacity. While I can 
speak from a direct involvement in the development of bubble chambers and 
measuring machines, and in the physics done with those tools, my relationship 
to our computer progamming efforts is largely that of an amazed spectator. 
We were most fortunate that in 1956 Frank Solmitz elected to join our group. 
Although the rest of the group thought of themselves as experimental physicists, 
Solmitz had been trained as a theorist, and had shown great aptitude in the 
development of statistical methods of evaluating experimental data. When 
he saw that our first Franckenstein was about to operate, and no computer 
programs were ready to handle the data it would generate, he immediately set 
out to remedy the situation. He wrote HYDRO, our first system program for 
use on the IBM 650 computer. In the succeeding twelve years he has con- 
tinued to carry the heavy responsibility for all our programming efforts. A 
major breakthrough in the analysis of bubble chamber events was made in the 
years 1957 through 1959. In this period, Solmitz and Art Rosenfeld, together 
with Horace Taft from Yale University and Jim Snyder from Illinois, wrote 
the first « fitting routine », GUTS, which was the core of our first « kinematics 
program, KICK ». To explain what KICK did, it is easiest to describe what 
physicists had to do before it was written. HYDRO and its successor, PANG, 
listed for each vertex the momentum and space angles of the tracks entering 
or leaving that vertex, together with the calculated errors in these measure- 
ments. A physicist would plot the angular coordinates on a stereographic 
projection of a unit sphere known as a Wolff-plot. If he was dealing with a 
three-track vertex — and that was all we could handle in those days — he 
would move the points on the sphere, within their errors, if possible, to make 
them coplanar. And of course he would simultaneously change the momentum 
values, within their errors, to insure that the momentum vector triangle closed, 
and energy was conserved. Since momentum is a vector quantity, the various 
conditions could be simultaneously satisfied only after the angles and the 
absolute values of the momenta had been changed a number of times in an 
iterative procedure. The end result was a more reliable set of momenta and 
angles, constrained to fit the conservation laws of energy and momentum. 
In a typical case, an experienced physicist could solve only a few Wolff-plot 
problems in a day. (Lynn Stevenson had written a specific program, COPLAN, 



Recent developments in particle physics 25 

that solved a particular problem of interest to him that was later handled by 
the more versatile GUTS.) 

GUTS was being written at a time when one higly respected visitor to the 
groups saw the large pile of PANG printout that had gone unanalysed beause 
so many of our group members were writing GUTS — a program that was 
planned to do the job automatically. Our visitor was very upset at what he 
told me was a « foolish deployment of our forces ». He said, « If you would 
only get all those people way from their program writing, and put them to 
work on Wolff-plots, we'd have the answer to some really important physics 
in a month or two ». I said I was sure we would end up with a lot more 
physics in the next years if my colleagues continued to write GUTS and 
KICK. I am sure that those who wrote these pioneering « fitting and kine- 
matics programs » were subjected to similar pressure. Everyone in the high- 
energy physics community has long been indebted to these farsighted men 
because they knew that what they were doing was right. KICK was soon 
developed so that it gave an overall fit to several interconnected vertices, 
with various hypothetical identities of the several tracks assumed in a series 
of attempts at a fit. The relationship beteen energy and momentum depends 
on mass, so a highly constrained fit can be obtained only if the particle 
responsible for each track is properly identified. If the degree of constraint 
is not so high, more than one « hypothesis » (set of track identifications) may 
give a fit, and the physicist must use his judgment in making the identification. 

As another example in this all-too-brief sketch of the computational aspects 
of our work, I will mention an important program, initiated by Art Rosenfeld 
and Ron Ross, that has removed much of the remaining drudgery from the 
bubble chamber physicists' life. SUMX is a program that can easily be in- 
structed to search quickly through large volumes of « kinematics program 
output », printing out summaries and tabulations of interesting data. (Like 
all our pioneering programs, SUMX was replaced by an improved and more 
versatile program— in this case, KIOWA. But I will continue to talk as 
though SUMX were still used.) A typical SUMX printout will be a com- 
puterprinted document 3 in.-thick, with hundreds of histograms, scatter 
plots, etc. 

Hundreds of histograms are similarly printed showing numbers of events 
with effective masses for many different combinations of particles, with 
various « cuts » on momentum transfer, etc. What all this amounts to is 
simply that a physicist is no longer rewarded for his ability in deciding what 
histograms he should tediously plot and then examine. He simply tells the 
computer to plot all histograms of any possible significance, and then flips 
the pages to see which ones have interesting features. 

One of my few real interactions with our programming effort came when 



26 L. W. Alvarez 

I suggested to Gerry Lynch the need for a program he wrote that is known 
as GAME. In my work as a nuclear physicist before World War II, I had 
often been skeptical of the significance of the « bumps » in histograms, to 
which importance was attached by their authors. I developed my own criteria 
for judging statistical significance, by plotting simulated histograms, assum- 
ing the curves to be smooth ; I drew several samples of « Monte Carlo distri- 
butions », using a table of random numbers as the generator of the samples. 
I usually found that my skepticism was well founded because the « faked » 
histograms showed as much structure as the published ones. There are of 
course many statistical tests designed to help one evaluate the reality of bumps 
in histograms, bu in my experience nothing is more convincing than an exam- 
ination of a set of simulated histograms from an assumed smooth distri- 
bution. 

GAME made it possible, with the aid of a few control cards, to generate 
a hundred histograms similar to those produced in any particular experiment. 
All would contain the same number of events as the real experiment, and 
would be based on a smooth curve through the experimental data. The 
standard procedure is to ask a group of physicists to leaf through the 100 
histograms — with the experimental histogram somewhere in the pile — and 
vote on the apparent significance of the statistical fluctuations that appear. 
The first time this was tried, the experimenter — who had felt confident that 
his bump was significant — did not know that his own histogram was in the 
pile, and did not pick it out as convincing; he picked out two of the computer- 
generated histograms as looking significant, and pronounced all other — 
including his own — as of no significance ! In view of this example, one can 
appreciate how many retractions of discovery claims have been avoided in 
our group by the liberal use of the GAME program. 

As a final example from our program library, I will mention FAKE, which, 
like SUMX, has been widely used by bubble chamber groups all over the 
world. FAKE, written by Gerry Lynch, generates simulated measurements of 
bubble chamber events to provide a method of testing the analysis programs 
to determine how frequently they arrive at an incorrect answer. 

Now that I have brought you up to date on our parallel developments of 
hardware and software (computer programs), I can tell you what rewards we 
have reaped, as physicists, from their use. The work we did with the 4 in. 
chamber at the 184 in. cyclotron and at the Bevatron cannot be dignified 
by the designation « experiments », but it did show examples of tz-\i-q decay 
and neutral strange-particle decay. The experiences we had in scanning the 
4 in. film merely whetted our appetite for the exciting physics we felt sure 
would be manifest in the 10 in. chamber, when it came into operation in 
Wilson Powell's big magnet. 



Recent developments in particle physics 27 

Robert Tripp joined the group in 1955, and as his first contribution to 
our program he designed a « separated beam » of negative K mesons that 
would stop in the 10 in. chamber. We had two different reasons for starting 
our bubble chamber physics program with observations of the behavior of 
K~ mesons stopping in hydrogen. The first reason involved physics: The 
behavior of stopping tz~ mesons in hydrogen had been shown by Panofsky 
and his co-workers [47] to be a most fruitful source of fundamental knowledge 
concerning particle physics. The second reason was of an engineering nature: 
Only one Bevatron « straight section » was available for use by physicists, 
and it was in constant use. In order not to interfere with other users, we 
decided to set the 10 in. chamber close to a curved section of the Bevatron, 
and use secondary particles, from an internal target, that penetrated the wall 
of the vacuum chamber and passed between neighboring iron blocks in the 
return yoke of the Bevatron magnet. This physical arrangement gave us 
negative particles (KT and it~ mesons) of a well-defined low momentum. 
By introducing an absorber into the beam, we brought the K~ mesons almost 
to rest, but allowed the lighter iz~ mesons to retain a major fraction of their 
original momentum. The Powell magnet provided a second bending that 
brought the K" mesons into the chamber, but kept the n~ mesons out. That 
was the theory of this first separated beam for bubble chamber use. But in 
practice, the chamber was filled with tracks of pions and muons, and we 
ended up with only one stopped K~ per roll of 400 stereo pairs. It is now 
common for experimenters to stop one million K" mesons in hydrogen, in 
a single experimental run, but the 137 K~ mesons we stopped in 1956 [48] 
gave us a remarkable preview of what has now been learned in the much 
longer exposures. We measured the relative branching of K~+p into 

S~ + 7i + :2 + 7r-:2; + ti :A + tt°. 

And in the process, we made a good measurement of the H° mass. We 
plotted the first decay curves for the 2+ and S _ hyperons, and we observed for 
the first time the interactions of S~ hyperons and protons at rest. We felt 
amply rewarded for our years of developmental work on bubble chambers 
by the very interesting observations we were now privileged to make. 

We had a most exciting experience at this time, that was the result of two 
circumstances that no longer obtain in bubble chamber physics. In the first 
place, we did all our own scanning of the photographic film. Such tasks are 
now carried out by professional scanner, who are carefully trained to recognize 
and record « interesting events ». We had no professional scanners at the time 
because we would not have known how to train them before this first film 
became available. And even if they had been trained, we would not have let 



28 L. W. Alvarez 

them look at the film— we found it so completely absorbing that there was 
always someone standing behind a person using one of our few film viewers, 
ready to take over when the first person's eyes tired. The second circumstance 
that made possible the accidental discovery I am about to describe was the 
very poor quality of our separated K~ beam— by modern standards. Most of 
the tracks we observed were made by negative pions or muons, but we also 
saw many positively charged particles— protons, pions and muons. 

At first we kept no records of any events except those involving strange 
particles; we would look quickly at each frame in turn, and shift to the next 
one if no « interesting event » showed up. In doing this scanning, we saw 
many examples of 7c+-[j+-e + decays, usually from a pion at rest, and we soon 
learned about how long to expect the (a+ track to be— about 1 cm. I did my 
scanning on a stereo viewer, so I probably had a better feeling for the length 
of a ji+ track in space than did my colleagues, who looked at two projections 
of the stero views, sequentially. Don Gow, Hugh Bradner and I often scanned 
at the same time, and we showed each other whatever interesting events 
came into view. Each of us showed the others examples of what we thought 
was an unusual decay scheme: 7t-->(jr->e- The decay of a [ir at rest into 
an e~ in hydrogen, was expected from the early observations by Conversi 
et al. [3], but Panofsky et al. [47] had shown that a tz~ meson could not decay 
at rest in hydrogen. Our first explanation for our observations was simply 
that the pion had decayed just before stopping. But we gradually became 
convinced that this explanation really did not fit the facts. There were too 
many muons tracks of about the same length, and none that were appreciably 
longer or shorter, as the decay-in-flight hypothesis would predict. We now 
began to keep records of these « anomalous decays », as we still called them, 
and we found occasional examples in which the muon was horizontal in the 
chamber, so its length could be measured. (We had as yet no way of recon- 
structing tracks in space from two stereo views.) By comparing the measured 
length of the negative muon track with that of its more normal positive counter- 
part, we estimated that the negative muons had an energy of 5.4 MeV, 
rather than the well-known positive muon energy (from positive pion decay 
at rest) of 4. 1 MeV. This confirmed our earlier suspicion that the long pri- 
mary negative track could not be that of a pion, but it left us just as much 
in the dark as to the nature of the primary. 

After these observations had been made, I gave a seminar describing what 
we had observed, and suggesting that the primary might be a previously 
unknown weakly interacting particle, heavier than the pion, that decayed into 
a muon and a neutral particle, either neutrino or photon. We had just made 
the surprising observation, shown in Fig. 10, that there was often a gap, meas- 
ured in millimeters, between the end of the primary and the beginning of 



Recent developments hi particle physics 



29 







k< 


\ 




• r±~ -4PW ■-*. 


N '; 1 \ 


. 


V 


V ■ * I 


^t ^ f: i 


. '1 I \ *.- 

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s * ■ ■ V 


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Fig. 10. - Muon catalysis (with gap). 



the secondary. This finding suggested diffusion by a ratlier long-lived negative 
particle that orbited around and neutralized one of the protons in the liquid 
hydrogen. We had missed many tracks with these « gaps » because no one 
has seen such a thing before; we simply ignored such track configurations by 
subconsciously assuming that they were unassociated events in a badly clut- 
tered bubble chamber. 

One evening, one of the members of our research team, Harold Ticho from 
our Los Angeles campus, was dining with Jack Crawford, a Berkeley astro- 
physicist he had known when they were students together. They discussed our 
observations at some length, and Crawford suggested the possibility that a 
fusion reaction might somehow be responsible for the phenomenon. They cal- 



30 L. W. Alvarez 

culated the energy released in several such reactions, and found that it agreed 
with experiment if a stopped muon were to be binding together a proton 
and a deuteron into an HD fir-molecular ion. In such a « mulecule » the 
proton and deuteron would be brought into such close proximity for such 
a long time that they would fuse into 3 He, and could deliver their fusion 
energy to the muon by the process of internal conversion. However, they 
could not think of any mechanism that would make the reaction happen so 
often — the fraction of deuterons in liquid hydrogen is only 1 in 5000. They 
had, however, correctly identified the reaction, but a key ingredient in the 
theoretical explanation was still missing. 

The next day, when we had all accepted the idea that stopped muons were 
catalyzing the fusion of protons and deuterons, our whole group paid a visit 
to Edward Teller, at his home. After a short period of introduction to the 
observations and to the proposed fusion reaction, he explained the high 
probability of the reaction as follows : the stopped muon radiated its way into 
the lowest Bohr orbit around a proton. The resulting muonic hydrogen atom, 
P(jr, then had many of the properties of a neutron, and could diffuse freely 
through the liquid hydrogen. When it came close to the deuteron in an HD 
molecule, the muon would transfer to the deuteron, because the ground 
state of the yrd atom is lower than that of the (orp atom, in consequence of 
« reduced mass » effect. The new « heavy neutron » dyr might then recoil 
some distance as a result of the exchange reaction, thus explaining the « gap ». 
The final stage of capture of a proton into a pdyc molecular ion was also 
energetically favorable, so a proton and deuteron could now be confined 
close enough together by the heavy negative muon to fuse into a 3 He nucleus 
plus the energy given to the internally converted muon. 

We had a short but exhilirating experience when we thought we had solved 
all of the fuel problems of mankind for the rest of time. A few hasty calcula- 
tions indicated that in liquid HD a single negative muon would catalyze enough 
fusion reactions before it decayed to supply the energy to operate an accelerator 
to produce more muons, with energy left over after making the liquid HD 
from sea water. While everyone else had been trying to solve this problem by 
heating hydrogen plasmas to millions of degrees, we had apparently stumbled 
on the solution, involving very low temperatures instead. But soon, more 
realistic estimates showed that we were off the mark by several orders of 
magnitude — a « near miss » in this kind of physics ! 

Just before we published our results [49], we learned that the « [j.-catalysis » 
reaction had been proposed in 1947 by Frank [50] as an alternative explana- 
tion of what Powell et al. had assumed (correctly) to be the decay of tt+ to (a+. 
Frank suggested that it might be the reaction we had just seen in liquid 
hydrogen, starting with a yr, rather than with a tz+. Zel'dovitch [51] had 



Recent developments in particle physics 



31 



extended the ideas of Frank concerning this reaction, but because their 
papers were not known to anyone in Berkeley, we had a great deal of personal 
pleasure that wc otherwise would have missed. 

I will conclude this episode by noting that we immediately increased the 
deuterium concentration in our liquid hydrogen and observed the expected, 
increase in fusion reactions, and saw two examples of successive catalyses by 
a single rnuon (Fig. II). We also observed the catalysis of D+D— > a H | ^H 
in pure liquid deuterium. 




Fig. II, - Double muon catalysis. 



32 



L. W. Alvarez 



A few months after we had announced our a-catalysis results, the world of 
particle physics was shaken by the discovery that parity was not conserved 
in beta decay. Madame Wu and her collaborators [52], acting on a suggestion 
by Lee and Yang [53], showed that the p rays from the decay of oriented 
6ft Co nuclei were emitted preferentially in a direction opposite to that of the 
spin. Lee and Yang suggested that parity nonconservation might also manifest 
itself in the weak decay of the A hyperon into a proton plus a negative pion. 
Crawford et at. had moved the 10 in. chamber into a negative pion beam, 
and were analysing a large sample of A*S from associated production events. 
They looked for an « up-down asymmetry » in the emission of pions from 
A'Sj relative to the « normal to the production plane », as suggested by Lee 
and Yang. As a result, they had the pleasure of being the first to observe 
parity nonconservation in the decay of hyperons [54]. 






Fig. 12. - K~ beam in 72 in. bubble chamber, a) No spectrometers on; b) one 
spectrometer on; c) two spectrometers on. 



Recent developments in particle physics 33 

In the winter of 1958, the 15 in. chamber had completed its engineering 
test run as a prototype for the 72 in. chamber, and was operating for the 
first time as a physics instrument. Harold Ticho, Bud Good and Philippe 
Eberhard [55] had designed and built the first separated beam of K~ mesons 
with a momentum of more than 1 GeV/c. Figure 12 shows the appearance 
of a bubble chamber when such a beam is passed through it, and when one 
or both of the electrostatic separators are turned off. The ingenuity which 
has been brought to bear on the problem of beam separation, largely by 
Ticho and Murray, is difficult to imagine, and its importance to the success 
of our program cannot be overstimated [55]. Joe Murray has recently joined 
the Stanford Linear Accelerator Center, where he has in a short period of 
time built a very successful radiofrequency-separated K beam and a back- 
scattered laser beam. 

The first problem we attacked with the 15 in. chamber was that of the H°. 
Gell-Mann had predicted that the S~ was one member of an /-spin doublet, 
with strangeness minus 2. The predicted partner of the H~ would be a neutral 
hyperon that decayed into a A and a t? — both neutral particles that would, 
like the H°, leave no track in the bubble chamber. A few years earlier, as an 
after-dinner speaker at a physics conference, Victor Weisskopf had « brought 
down the house » by exhibiting an absolutely blank cloud chamber photo- 
graph, and saying that it represented proof of the decay of a new neutral 
particle into two other neutral particles ! And now we were seriously planning 
to do what had been considered patently ridiculous only a few years earlier. 

According to the Gell-Mann and Nishijima strangeness rules, the H° should 
be seen in the reaction 

K~+p^H° + K° 

\ \ 

A+TC° TC _ +TC + 
I 

In the one example of this reaction that we observed, Fig. 13, the charged 
pions from the decay of the neutral K° yielded a measurement of the energy 
and direction of the unobserved K°. Through the conservation laws of energy 
and momentum (plus a measurement of the momentum of the interacting K~ 
track) we could calculate the mass of the coproduced S° hyperon plus its 
velocity and direction of motion. Similarly, measurements of the if and 
proton gave the energy and direction of motion of the unobserved A, and 
proved that it did not come directly from the point at which the K~ meson 
interacted with the proton. The calculated flight path of the A intersected 
the calculated flight path of the H°, and the angle of intersection of the two 



34 



L. W. Alvarez 




Fig, 13. - Production and decay of a neulrat cascade hyperon (E°). 



unobserved but calculated tracks gave a confirming measurement of the 
mass of the E 1 hyperon, and proved that it decayed into a A plus a - 
This single hard-won event was a sort of tour de force that demonstrated 
clearly the power of the liquid hydrogen bubble chamber plus its associated 
data-analysis techniques. 

Although only one 2° was observed in the short time the 15 in. chamber 
was in the separated K beam, large numbers of events showing strange- 
particle production were available for study. The Franckcnsteins were kept 
busy around the clock measuring these events, and those of us who had 
helped to build and maintain the beam now concentrated our attention on 
the analysis of these reactions. The most copious of the simple « topologies » 
was K"p -*■ two charged prongs plus a neutral V-partielc. According to the 



Recent developments in particle physics 35 

strangeness rules, this topology could represent either 

K"+p-^A+7r++7r- 
I 

71 +p 

or 

K~-> p -> K°+p+^ - 

71 -\-ll + 

The kinematics program, KICK, was now available to distinguish between 
these two reactions, and to eliminate those examples of the same topology in 
which an unobserved 7i° was produced at the first vertex. SUMX had not yet 
been written, so the labor of plotting histograms was assumed by the two very 
able graduate students who has been associated with the K~ beam and its 
exposure to the 15 in. chamber since its planning stages: Stanley Wojcicki 
and Bill Graziano. They first concentrated their attention on the energies of 
the charged pions from the production vertex in the first of the two reactions 
listed above. Since there were three particles produced at the vertex — a 
charged pion of each sign plus a A — one expected to find the energies of each 
of the three particles distributed in a smooth and calculable way from a mini- 
mum value to a maximum value. The calculated curve is known in particle 
physics at the « phase-space distribution ». The decay of a t meson into three 
charged pions was a well-known « three-particle reaction » in which the 
dictates of phase space were rather precisely followed. 

But when Wojcicki and Graziano finished transcribing their data from 
KICK printout into histograms, they found that phase-space distributions 
were poor approximations to what they observed. Figure 14 shows the 
distribution of energy of both positive and negative mesons, together with 
the corresponding « Dalitz plot », which Richard Dalitz [56] had originated 
to elucidate the « t-6 puzzle », which had in turn led to Lee and Yang's parity- 
nonconservation hypothesis. 

The peaked departure from a phase-space distribution had been observed 
only once before in particle physics, where it had distinguished the reaction 
p+p-^ 7i + +d from the « three-body reaction » p+p-» 7r + +p+n. (Although 
no new particles were discovered in these reactions, they did contribute to our 
knowledge of the spin of the pion [57]). But such a peaking had been observed 
in the earliest days of experimentation in the artificial disintegration of nuclei, 
and its explanation was known from that time. Oliphant and Rutherford [58] 
observed the reaction p+ n B->3 4 He. This is a three-body reaction, and the 
energies of the a particles had a phase-space-like distribution except for the 
fact that there was a sharp spike in the energy distribution at the highest 



36 



L. W. Alvarez 




40 80 120 160 200 240 280 320 360400 
TV+dVlev) 



Fig. 14. - Discovery of the Y*(1385) (see text for explanation). 



a-particle energy. This was quickly and properly attributed [58] to the reaction 



p+ n B^ 8 Be+ 4 He 



l He+ 4 He 



In other words, some of the reaction proceeded via a two-body reaction, 
in which one a particle recoiled with unique energy against a quasistable 
8 Be nucleus. But the 8 Be nucleus was itself unstable, coming apart in 10~ 16 s 
into two a particles of low relative energy. The proof of the fleeting existence 
of 8 Be was the peak in the high-energy a-particle distribution, showing that 
initially only two particles, 8 Be and 4 He, participated in the reaction. 

The peaks seen in Fig. 14 were thus a proof that the Tii recoiled against 
a combination of A+tt^ that had a unique mass, broadened by the effects of 
the uncertainty principle. The mass of the Arc combination was easily calculable 



Recent developments in particle physics 



37 



as 1382 MeV, and the /-spin of the system was obviously 1, since the /-spin 
of the A is 0, and the /-spin of the 7i is 1. This was then the discovery of the 
first « strange resonance », the Yi(1385). Although the famous Fermi 3,3 
resonance had been known for years, and although other resonances in the 
-nr nucleon system had since shown up in total cross-section experiments at 




600 700 800 900 

Mass of K IT system ( Mev ) 



Fig. 15. - Discovery of the K*(890). 



38 



L. W. Alvarez 



Brookhaven and Berkeley, CalTech and Cornell [59] the impact of the Y* 
resonance on the thinking of particle physicists was quite different — the Y£ 
really acted like a new particle, and not simply as a resonance in a cross- 
section. 



25 



20 



15 



"I 1 r 




* + 



-96events 




10 



5- 



<i 



Typical mass 
uncertainty 



I25events 
96 events 








1400 1500 

M°(Mev) 



1600 



Fig. 16. - Discovery of the Y*(1405). 



We announced the Yj at the 1960 Rochester High Energy Physics Con- 
ference [60], and the hunt for more short-lived particles began in earnest. The 
same team from our bubble chamber group that had found the Yi(1385) 



Recent developments in particle physics 



39 



now found two other strange resonances before the end of 1960 — the 
K*(890) [61], and the Y£(1405) [62]. 

Although the authors of these three papers have for years been referred to 
as « Alston et al. », I think that on this occasion it is proper that the full list 
be named explicitly. In addition to Margaret Alston (now Margaret Garnjost) 
and Luis W. Alvarez, and still in alphabetical order, the authors are : Philippe 
Eberhard, Myron L. Good, William Graziano, Harold K. Ticho, and Stanley 
G. Wojcicki. 

Figures 15 and 16 show the histograms from the papers announcing these 
two new particles; the K* was the first example of a « boson resonance » 
found by any technique. Instead of plotting these histograms against the 
energy of one particle, we introduced the now universally accepted technique 
of plotting them against the effective mass of the composite system: S+tt for 
the Yo(1405) and K+tt for the K*(890). Figure 17 shows the present state 
of the art relative to the K*(890) ; there is essentially no phase-space background 
in this histogram, and the width of the resonance is clearly measurable to 
give the lifetime of the resonant state via the uncertainty principle. 




700 800_ 900 1000 1100 1200 

Mass of kV- system (MeV) 



Fig. 17. - Present day K*(890). 



These three earliest examples of strange-particle resonances all had lifetimes 
of the oder of 10~ 23 s, so the particle all decayed before they could traverse 
more than a few nuclear radii. No one had foreseen that the bubble chamber 
could be used to investigate particles with such short lives; our chambers 



40 



L. W. Alvarez 



had been designed to investigate the strange particles with lifetimes of lO -10 s— 
lO 13 times as long. 

In the summer of 1959, the 72 in. chamber was used in its first planned 
physics experiment. Lynn Stevenson and Philippe Eberhard designed and 
constructed a separated beam of about l.6GeV/c autiprotons, and a quick 
scan of the pictures showed the now famous first example of antilambda 
production, via the reaction 



PH P 



■A + 



A 




Fig, 18. - First production of anti-lambda. 



Recent developments in particle physics 41 

Figure 18 shows this photograph, with the antiproton from the antilambda 
decay annihilating in a four-pion event. I believe that everyone who attended 
the 1959 High Energy Physics Conference in Kiev will remember the showing 
of this photograph — the first interesting event from the newly operating 72 in. 
chamber. 

Hofstadter's classic experiments on the scattering of high energy electrons 
by protons and neutrons [63] showed for the first time how the electric charge 
was distributed throughout the nucleons. The theoretical interpretation of the 
experimental results [64] required the existence of two new particles, the 
vector mesons now known as the co and the p. The adjective « vector » simply 
means that these two mesons have one unit of spin, rather than zero, as the 
ordinary tz and K mesons have. The co was postulated to have /-spin = 0. 
and the p to have /-spin = 1 ; the co would therefore exist only in the neu- 
tral state, while the p would occur in the +, — , and charged states. 

Many experimentalists, using a number of techniques, set out to find these 
important particles, whose masses were only roughly predicted. The first 
success came to Bogdan Maglic, a visitor to our group, who analysed film from 
the 72 in. chamber's antiproton exposure. He made the important decision 
to concentrate his attention on proton-antiproton annihilations into five 
pions — two negative, two positive, and one neutral. KICK gave him a selected 
sample of such events; the tracks of the 7i° could not be seen, of course, 
but the constraints of the conservation laws permitted its energy and direction 
to be computed. Maglic then plotted a histogram of the effective mass of 
all neutral three-pion combinations. There were four such neutral combinations 
for each event; the neutral pion was taken each time together with all four 
possible pairs of oppositely charged pions. SUMX was just beginning to 
work, and still had bugs in it, so the preparation of the histogram was a 
very tedious and time-consuming chore, but as it slowly emerged, Maglic 
had the thrill of seing a bump appear in the side of his phase-space distri- 
bution. Figure 19 shows the peak that signaled the discovery of the very 
important co meson. 

Although Bogdan Magli6 originated the plan for this search, and pushed 
through the measurements by himself, he graciously insisted that the paper 
announcing his discovery [65] should be co-authored by three of us who 
had developed the chamber, the beam, and the analysis program that made 
it possible. 

The p meson is the only one from this exciting period in the development 
of particle physics whose discovery cannot be assigned uniquely. In our 
group, the two Frankensteins were being used full time on problems that the 
senior members felt had higher priority. But a team of junior physicists and 
graduate students, Anderson et al. [66], found that they could make accurate 



42 



L. W. Alvarez 



2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 



Q. 



E 

3 




0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 1.3 1.4 1.5 1.6 1.7 1.8 IJ9 2.0 2.1 2.2 

Effective mass , M 3 (Bev) 



Fig. 19. - Discovery of the <o meson. 



Recent developments in particle physics 



43 



enough measurements direcly on the scanning tables to accompliesh a « Chew- 
Low extrapolation ». Chew and Low had described a rather complicated 
procedure to look for the predicted dipion resonance now known as the p 
meson. Figure 20 shows the results of this work, which convinced me that 
the p existed and had its predicted spin of 1 . The mass of the p was given 
as about 650 MeV, rather than its now accepted value of 765 MeV. (This 
low value is now explained in terms of the extreme width of the p resonance.) 
The evidence for the p seemed to me even more convincing than the early 
evidence Fermi and his co-workers produced in favor of the famous 3, 3 
pion-nucleon resonance. 



Fit 



-. 400 

E 
Is 300 

« 200 

100 



( Constroinf) 
♦ 

Lin. Quad. Quod. Quod. Lin. Lin. Lin. Lin. 

X' : Prob(%) 50 4 50 75 93 90 50 65 

Lin. Lin. 

0.8% 35% 

No. of events: 74 127 176 170 197 178 173 105 




( 2J + I) 47T*. for J=l 



■4 151 Quod . 22 25 28 

CU 2 (m7r 2 ) 
Fig. 20. - First evidence for the p meson. 



But one of the unwritten laws of physics is that one really has not made a 
discovery until he has convinced his peers that he has done so. We had just 
persuaded high energy physicists that the way to find new particles was to 
look for bumps on effective-mass histograms, and some of them were therefore 
unimpressed by the Chew-Low demonstration of the p. Fortunately, Walker 
and his collaborators [67] at Wisconsin soon produced an effective-mass 
ideogram with a convincing bump at 765 MeV, and they are therefore most 
often listed as the disco veres of the p. 

Ernest Lawrence very early established the tradition that his laboratory 
would share its resources with others outside its walls. He supplied short- 
lived radioactive materials to scientists in all departments at Berkeley, and 



44 L. W. Alvarez 

he sent longer-lived samples to laboratories throughout the world. The first 
artificially created element, technetium, was found by Perrier and Segre [68], 
who did their work in Palermo, Sicily. They analysed the racioactivity in a 
molybdenum deflector strip from the Berkeley 28 in. cyclotron that had been 
bombarded for many months by 6 MeV deuterons. 

We followed Ernest Lawrence's example, and thus participated vicariously 
in a number of important discoveries of new particles. The first was the v\ 
found at Johns Hopkins, by a group headed by Aihud Pevsner [69]. They 
analysed film from the 72 in. chamber, and found the yj with a mass of 
550 MeV, decaying into 7t+tc~7i . Within a few weeks of the discovery of 
the •/), Rosenfeld and his co-workers [70] at Berkeley, who had independently 
observed the yj, showed quite unexpectedly that /-spin was not conserved in 
its decay. Figure 21 shows the present state of the art with respect to the o> 
and 7) mesons; the strengths of their signatures in this single histogram is 
in marked contrast to their first appearance in 72 in. bubble chamber experi- 
ments. 

In the short interval of time between the first and second publications on 
the y], the discovery of the Yq(1520) was announced by Ferro-Luzzi, Tripp, 
and Watson [71], using a new and elegant method. Bob Tripp has continued 
to be a leader in the application of powerful methods of analysis to the study 
of the new particles. 

The discovery of the S*(1530) hyperon was accomplished in Los Angeles 
by Ticho and his associates [72], using 72 in. bubble chamber film. Harold 
Ticho had spent most of his time in Berkeley for several years, working tirelessly 
on every phase of our work, and many of his colleagues had helped prepare 
the high-energy separated K~ beam for what came to be known as the K72 
experiment. The UCLA group analysed the two highest-momentum K~ 
exposures in the 72 in. chamber, and found the E*(1530) just in time to report 
it at the 1962 High Energy Physics Conference in Geneva. (Confirming 
evidence for this resonance soon came from Brookhaven [73]). 

Murray Gell-Mann had recently enunciated his important ideas concern- 
ing the « Eightfold Way » [74], but his paper had not generated the interest 
it deserved. It was soon learned that Ne'eman had published the same sug- 
gestions, independently [75]. 

The announcement of the S*(1530) fitted exactly with their predictions 
of the mass and other properties of that particle. One of their suggestions was 
that four /-spin multiplets, all with the same spin and parity, would exist in 
a « decuplet » with a mass spectrum of « lines » showing an equal spacing. 
They put the Fermi 3, 3 resonance as the lowest mass member, at 1238 MeV. 
The second member was the Yi(1385), so the third member should have a 
mass of (1385) + (1385— 1238) = 1532. The strangeness and the multiplicity 



Recent developments in particle physics 



45 





1000 




800 




600 


> 




0) 


400 


m 




V. 


200 


c 
> 





o 






600 


jQ 

E 




IS 


400 



200 







(a) 



I.I - 1.8 BeV/c 




.5 BeV/c 



ft 




400 



600 



800 



+ _-_ o 



M (7r + 7r"7r ) 



MeV 



000 



Fig. 21. - Present day histogram showing w and v\ mesons. 



of each member of the spectrum was predicted to drop 1 unit per member, 
so the E*(1530) fitted their predictions completely. It was then a matter of 
simple arithmetic to set the mass, the strangeness, and the charge of the 



46 L. W. Alvarez 

final member — the Q~. The realization that there was now a workable theory 
in particle physics was probably the high point of the 1962 International 
Conference on High Energy Physics. 

Since the second and third members of the series — the ones that permitted 
the prediction of the properties of the D~ to be made — had come out of our 
bubble chamber, it was a matter of great disappointment to us that the 
Bevatron energy was insufficient to permit us to look for the Q,~. Its widely 
acclaimed discovery [76] had to wait almost two years, until the 80 in. chamber 
at Brookhaven came into operation. 

Since the name of the Q. had been picked to indicate that it was the last 
of the particles, the mention of its discovery is a logical point at which to 
conclude this lecture. I will do so, but not because the discovery of the Q. 
signaled the end of what is sometimes called the population explosion in 
particle physics — the latest list [77] contains between 70 and 100 particle 
multiplets, depending upon the degree of certainty one demands before 
« certification ». My reason for stopping at this point is simply that I have 
discussed most of the particles found by 1962 — the ones that were used by 
Gell-Mann and Ne'eman to formulate their SU 3 theories — and things became 
much too involved after that time. So many groups were then in the « bump- 
hunting business » that most discoveries of new resonances were made simul- 
taneously in two or more laboratories. 

I am sorry that I have neither the time nor the ability to tell you of the 
great beauty and the power that has been brought to particle physics by 
our theoretical friends. But I hope that before long, you will hear it directly 
from them. 

In conclusion, I would like to apologize to those of my colleagues and my 
friends in other laboratories, whose important work could not be mentioned 
because of time limitations. By making my published lecture longer than the 
oral presentantion, I have reduced the number of apologies that are necessary, 
but unfortunately I could not completely eliminate such debts. 



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48 L. W. Alvarez 

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references in R. F. Peierls: Phys. Rev., 118, 325 (1959). 
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Recent developments in particle physics 49 

[69J A. Pevsner, R. Kraemer, M. Nussbaum, C. Richardson, P. Schlein, R. Strand, 
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Weak Interactions 
and the Breaking of Hadron Symmetries. 

N. Cabibbo 

Istituto di Fisica delVUniversita - Roma 
Istituto Nazionale di Fisica Nucleare - Sezione di Roma 

L. Maiani 

Laboratori di Fisica, Istituto Superiore di Sanitd - Roma 
Istituto Nazionale di Fisica Nucleare, Sottosezione Sanitd - Roma 



Introduction. 

In a recent paper [1] we have proposed a dynamical relation among weak, 
strong, and e.m. interactions, on the basis of a self-consistency condition 
which requires that the description of hadron physics remains unchanged 
by the effect of leading weak corrections and e.m. tadpole effects. The main 
results of this approach were a relation between the weak interaction angle 6 
and the symmetry breaking parameters, and the appearance of a nonelectro- 
magnetic breaking of isotopic spin. 

The presentation in ref. [1] was centered on the application of the self- 
consistency condition itself. This condition is, to a certain extent, arbitrary 
at the present stage of the theory and a deeper understanding of the whole 
subject is certainly needed to acquire more confidence in its validity. In this 
paper we present a review of the whole theory putting a particular emphasis 
on those aspects which do not require explicitly the use of the self-consistency 
condition. These aspects are interesting on their own, and pose many open 
problems to future investigations. 

In Sect. 1 we give an outline of hadron symmetries and their connection 
to weak interactions. Section 2 contains a phenomenological analysis of 
symmetry breaking. On this point we follow the analysis of Gell-Mann 
et al. [2] and of Glashow and Weinberg [3], and in addition we discuss the 
necessity, at a phenomenological level, of introducing a nonelectromagnetic 
isospin breaking. In Sect. 3 we discuss attempts to understand the structure 



Weak interactions and the breaking of hadron symmetries 51 

of symmetry breaking in terms of a purely strong interaction dynamics. 
In this connection we give a simple derivation of a result due to Michel and 
Radicati [4], indicating that the patterns of symmetry breaking preferred by 
strong interaction bootstrap are those which reduce SU Z (x) SU 3 to either 
SU 3 or SU 2 (x) SU 2 . Section 4 is devoted to an analysis of the consequences 
of the non-e.m.-isospin breaking previously introduced with respect to 
y] -> 3tc decay and to the mass splittings within isospin multiplets. 

In Sect. 5 we study the possible strong effects of weak interactions. In 
particular we show that if the explicit strong breaking of SU 3 (x) SU 3 trans- 
forms as a (3, 3) + (3, 3) representation, these corrections do not produce 
a breaking of parity and strangeness at a strong level. We also show that 
this is not true if the explicit breaking transforms as a (1, 8) + (8, 1). 

Finally, in Sect. 6, we review our self-consistency requirement and its 
consequences. 

1. - Investigations on symmetries in elementary particle physics have been 
pursued with an ever increasing degree of effort and sophistication in the 
last decade. 

The well-established isospin symmetry acquired a new dimension with 
the Conservation of Vector Current (CVC) hypothesis of Feynman and Gell- 
Mann [5], which identified the charged isospin currents with a part of the 
vector current appearing in the weak coupling of leptons to hadrons. The 
neutral currents related to I z and to the hypercharge Y were already identified 
as components of the electromagnetic current. The CVC hypothesis thus 
accomplished the program of giving a physical role to the generators of the 
whole SU 2 ®U 1Y , which at that time represented (apart from baryonic number 
and discrete symmetries) the full invariance group of strong interactions. 

It was then natural to try the opposite approach, namely to give a sym- 
metry role to all currents with a physical meaning, and in particular to the 
axial and the strangeness changing currents (both axial and vector) appear- 
ing in (3-decays. 

Consideration of the strangeness nonchanging axial current led to the 
extension of the SU 2 group of isospin into the chiral SU 2 (x) SU 2 group. This 
interpretation of the axial current has been made possible by the use of an 
entirely new concept, i.e., that of a dynamically broken symmetry, introduced 
by Nambu and Jona-Lasinio [6]. The chiral SU 2 (x) SU 2 symmetry, in fact, 
is not realized in the usual way, as this would require all hadrons to appear 
in degenerate parity doublets. Baryons not appearing in parity doublets 
should have a vanishing mass, a possibility even further from physical reality. 
That chiral SU 2 (x) SU 2 symmetry can nevertheless be realized is made plau- 
sible by the following argument: A continuous symmetry group implies 



52 N. Cabibbo and L. Maiani 

the existence of a set of operators commuting with the Hamiltonian. One 
of these operators, when acting on a single particle state, should turn it into 
states of the same mass. 

In the usual realization of a symmetry, these new states are also one 
particle states, and this requires particles to appear in degenerate multiplets, 
which form a basis for a representation of the group in question. The gen- 
erators of a dynamically broken symmetry turn instead single particle into 
multiparticle states, and in order for these states to have the same mass as 
the original one, massless bosons must appear. In the case of chiral SU 2 (x) 
(x) SU 2 , these bosons are identified with the pions, and the symmetry is exact 
to the extent that one can neglect the pion mass. An axial generator turns 
then, e.g., a single nucleon state into « one nucleon plus many pions » states 
and this avoids the parity doubling of nucleon states. It is clear that the 
hypothesis of chiral SU 2 ® SU 2 symmetry for strong interactions does not 
lead to new predictions on hadron spectrum, but provides a powerful tool 
for relating processes involving low energy (soft) pions. The exploitation of 
soft pion theorems, pioneered by Nambu, has received a great amount of 
attention in the last few years, yielding numerous results in good agreement 
with experiments [7]. The same results have also been obtained by an equiv- 
alent approach based on the hypothesis of Partial Conservation of the 
Axial Current (PCAC), introduced by Gell-Mann and Levy [8]. 

Only after the discovery of SU 3 as an approximate symmetry of hadrons, 
it has been possible to complete the program of giving a symmetry inter- 
pretation for the strangeness changing weak current. Since SU 3 is a symmetry 
of the normal kind, akin to isotopic spin, it allows a classification of hadrons 
in supermultiplets comprising multiplets of different isospin and strangeness. 

The two extensions of SU 2 , i.e., chiral SU 2 ® SU 2 and SU 3 , both well 
supported by experimental facts, can only coexist if they are subgroups of 
a larger symmetry. The simplest possibility for the larger group is a chiral 
SU 3 (x) SU 3 . Associated with this larger group is a set of eight vector and 
eight pseudovector currents, Vj^x) and Afa) (i= 1,...,8). One can com- 
bine these currents into two sets: 

0) Jl-Vl + A^, 

(2) Jp = V n A^ , 

whose charges generate two commuting SU 3 . 

It has been conjectured [9] that the weak current of hadrons is a com- 
bination of these according to: 

(3) J?* = cosd(J l _|_ ijt) + sin (/ 4 + /7 5) f 



Weak interactions and the breaking of hadron symmetries 53 

being a new, universal constant. Equation (3) has been used to give a 
simple description of all [3-decay processes, which has been fully confirmed 
by the existing experimental data [10]. In particular has been determined 
to be ~0.22. 

The structure of the theory can be presented in a very simple way, using 
the language of the quark model. In this model all hadrons are bound states 
of the three quarks p, n, X, and their antiparticles. The quarks themselves 
transform as the basic representation 3 of SU 2 . The symmetry currents /* 
and J^ have the simple expression: 



(4) 






where li (7=1,..., 8) are the eight Gell-Mann matrices. The hadronic 
weak current, eq. (3), has the simple form: 

(5) /; eak =py,(l + y 5 )[cos0w + sin0A] = fy^l + y s )X+y> , 

where 

(0 cos0 sin0 N 




In this picture the meaning of the parameter is particularly transparent. 
The weak current has a specially simple form in terms of the three fields: 
p, n' = cosdn + sinflA, A' = — sin0« + cos0A: 

(7) Jjr k =PY l &+YJn'- 

Thus is seen to be the angle between the frame p, n, A and p, n', X' chosen 
in the quark internal space by the SU 3 breaking and by the weak interactions 
respectively. The current given by (5) and (7) displays exactly the same 
structure as the (ev e ), (fxv^) pieces of the leptonic current, thus obeying the 
requirement of universality of weak interactions. The universality principle 
can be given a more abstract formulation in terms of current commutators [11], 
and can be proven to be satisfied by eq. (3) without any reference to the 
quark model. 

su 2 ( 2 SU *® SU 3 

^ su 2 ®su 2 s 

Fig. 1. 



54 N. Cabibbo and L. Maiani 

2. - In the previous section we have outlined the emergence of a chain 
of higher symmetries for hadrons, from SU 2 (x) U lY to SU s (x)SU 3 , according 
to the pattern of Fig. 1 . We have also emphasized the deep connection between 
weak and e.m. interactions on one side and the symmetry structure of strong 
interactions on the other side. In this Section we will present a phenomeno- 
logical study of the breaking of these symmetries following the works of 
refs. [2] and [3]. This discussion will be based on the usual assumption that 
hadron physics can be described by a hierarchy of interactions, where the 
main features of hadronic processes are determined by strong interactions, 
with small e.m. and weak perturbation. This is in fact a very strong assumption, 
which will probably be abandoned in the future developments of the theory. 
As we shall see later, weak interactions can give rise to large effects, strictly 
interwoven with what one would call proper strong effects. 

With this proviso, let us write the hadron Lagrangian as: 

(8) &=&M+&e J n.+ &*. 

We shall also write j2? s = jS? + jSPj , where =^ is that part of the strong 
Lagrangian which is symmetric under the full SU s (x) SU 3 group. We will 
refer to <g x as the explicit breaking term of & s . In fact, even in the limit 
=2\ = and neglecting weak and e.m. effects, one can still have a dynamical 
breaking, as discussed in the previous section for the case of spontaneous 
breaking of SU 2 (g) SU 2 . In this case a particular realization of =^ would 
be characterized by a subgroup G of SU 3 (x) SU S , represented in the multi- 
plet structure of hadrons. To each of the SU 3 (g) SU 3 generators not belong- 
ing to G it would correspond a scalar or pseudoscalar massless boson. Some 
possibilities are given in Table I. Only the first one appears to be near the 
physical situation. In fact the hadron spectrum clearly exhibits the SU 3 
multiplet structure, and the eight pseudoscalar mesons are the least massive 
of known hadrons. 

Table I. - Some possibilities for the spontaneous breaking of SU 3 xSU 3 . K s indicates a 

scalar kappa meson. 

Multiplet Structure Massless Bosons 

SV» 7U, K, 7) 

Chiral SU 2 ®SU 2 K, t], K s 

SU 2 ® U 1T tz, K, ti, K s 



A satisfactory description of ST/ 3 (x) SU 3 breaking appears then to be one 
in which the symmetry of ^ is realized according to the first possibility of 
Table I. The explicit breaking term jSf x would then cause a departure from 



Weak interactions and the breaking of hadron symmetries 



55 



this situation, giving finite masses to the pseudoscalar mesons as well as mass 
splittings within SU S multiplets. This picture provides a rational basis to 
the observation that the masses of the pseudoscalar mesons are of the same 
order of magnitude as their mass splittings, and of mass splittings within 
other multiplets. 

The fact that the pion mass is an extremely small quantity on the hadron 
scale (m£ ^ 0.019 GeV 2 ) together with the considerations of Sect. 1 indicate 
that the structure of J^ is such as to give a relatively large breaking of 
SU s (g) SU 3 into SU 2 (x) SU 2 , and a much smaller breaking of chiral SU 2 (x) 
(x) SU 2 . The success of the Gell-Mann-Okubo mass formula moreover 
indicates that S£ x is a superposition of SU S singlets and octets. All these 
requirements are met if we assume that £P X transforms as a component of a 
(3, 3) ©(3, 3) representation of chiral SU Z ® SU 3 . 

The basis for such a representation is a set of nine scalar and nine pseudo- 
scalar densities a* and n 1 [12], obeying the equal-time commutation relations: 



[Vi,aj] = 

[Vi,7lj] = 
[Ai,7Tj] = 



ifijkGk , 
ifijk^k , 

idtjicTik , 
— idijkGk • 



In terms of these objects, (assuming parity and strangeness conservation) 
SP-l has the simple form: 

(9) Se x = e Q a + £ 8 cr 8 + e z a z . 

This picture has a simple transcription in quark language, where: 

and the breaking term £P X can be interpreted as a quark mass term: 



(10) 
where : 



(11) 



&\ — X PP + finn + yAA 



oc = l/3 £ o+^| + £3 ' 



fi« 



P= V ***+-!*-*• 



V3 



1h 



7= ^ £ »-Vf £8 



56 N. Cabibbo and L. Maiani 

The requirement that & x leads to a nearly exact SU 2 (x) SU 2 symmetry can 
be stated as saying that p and n masses must be much smaller than the X 
mass, i.e. 

or, equivalently, 

e 8 ?w — V2 e . 

Detailed phenomenological analyses have been carried out in refsJ[2] 
and [3], and indicate that: 

(12) e 8 / £o ^-1.25. 

In eq. (9) we have attributed to j2\ also a term proportional to a z which causes 
a breakdown of isospin symmetry, independent from the isospin breaking 
induced by the electromagnetic interaction J? e . m .. The necessity of such a 
term is indicated by two different theoretical results. The first one is a theorem 
due to Bell and Sutherland [13], according to which in the soft pion limit 
the isospin violating decay 7] -> 3tc cannot proceed through e.m. interaction. 
Bell and Sutherland have also shown that the observed decay rate is by at 
least two orders of magnitude larger than what could be obtained introducing 
the corrections for a finite pion mass. The second argument relies on a theorem 
by R. Dashen [14], which states that, if isospin breaking is of a purely e.m. 
origin the sum rule: 



2 k + — m K! 



mt+ — rrivo = ml+ — mi» 



should be verified up to terms of order e 2 e 8 and e 2 e . The failure of this sum 
rule, which reads: 

— 4000 MeV 2 = 1260 MeV 2 

has to be compared with the good agreement of the Coleman and Glashow [15] 
sum rule: 

E~—E = p — n+ E~—S + 

which is derived also neglecting terms of order e 2 e 8 . The problems arised 
by the theorems of Bell and Sutherland and of Dashen can both be solved 
by introducing a <r 3 term in J^ . Question arises whether one can fix the para- 
meter e 3 in such a way as to obtain a quantitative agreement in both cases. 
This question will be discussed in Sect. 4 where we will also analyze other 
possible consequences of the <r 3 interaction. 



Weak interactions and the breaking of hadron symmetries 57 

3. - We want now to consider the question whether the structure of 
symmetry breaking which we have studied at a phenomenological level, and 
in particular the values for the parameters e , e 8 , e 3 , can be understood in the 
framework of current theories of strong interactions. 

Our present understanding of strong interaction dynamics is based 
on the idea of bootstrap according to which the parameters of the theory 
should be determined by self-consistency conditions. The study of SU 3 
breaking within a bootstrap theory has been pioneered by R. Cutkosky [16], 
who found that solutions of the bootstrap are favored, where the breaking 
reduces SU 3 to exact SU 2 . Michel and Radicati [17] have produced an 
interesting geometrical insight into this result, and have recently ex- 
tended [4] their study to the breaking of SU 3 (g) SU 3 . We shall state here 
their result, and discuss its limitations and its implications for our program. 
In Appendix I we shall give a simple derivation of the Michel-Radicati 
theorem. 

Let us assume that the bootstrap condition can be expressed as a varia- 
tional principle: 

(13) 8G(e 3 ,e 8 ,e ) = 0. 

Equation (13) will have to be completed by some other requirement, to be 
called the « stability condition » which will select a solution of eq. (13) as 
the physically relevant one. 

We do not need to give here a specific form to these stability conditions; 
they could simply consist in the requirement that the stable solution be a 
minimum of G, or could be of a more complicated nature [18]. 

The result of Michel and Radicati then states that there exist always 
solutions of eq. (13) which correspond to: 

i) exact SU 3 (x) SU 3 (e 3 = e 8 = s — 0) ; 
ii) exact SU 3 (e 3 = e 8 = ; £ ^ 0) ; 
iii) exact chiral SU 2 ® SU 2 (e 3 =0; £ 8 = — V^ e o)- 

The significance of this result is limited by the consideration that one is 
not able to prove the nonexistence of other solutions which might be more 
« stable » than the previous ones. 

The Michel-Radicati theorem can be taken as an indication that strong 
interactions favor solutions of the kind i), ii), iii), which are different from 
the one observed (even if only slightly in the case of solution iii)). We obtain 
therefore from their result some support to the idea that weak and e.m. 
interactions play a nonnegligible role in determining the structure of the 
breaking. In order to prove the existence of solutions ii) and iii), Michel 



58 N. Cabibbo and L. Maiani 

and Radicati restrict the range of variation of the e's to the unit sphere: 

e 3 ~f~ £ 8 ~~t~ £ == 1 • 

As shown in Appendix I this restriction can be justified by the physical 
ipothesis that bootstrap equations are not able to fix the scale of hadron 
masses, i.e., that G has the scale invariance property: 

G{Ae z , Ae 8 , As ) = G(e 3 , e 8 , e ) . 

4. - Non-e.m. breaking of /-spin. 

In this section we describe possible physical consequence of a non-e.m. 
breaking of isospin, as embodied in the term e s a 3 of J§f x . 

Electromagnetism itself can give rise to corrections which imitate the 
effect of such a term. These corrections are the so-called tadpole contributions 
of the e.m. interactions. They arise from graphs of the kind shown in Fig. 2, 




| cr? i "* i^o 



Fig. 2. 



which correspond to the annihilation into the vacuum of scalar mesons 
associated with the a 3 , a 6 , and ff densities. We may separate this contribu- 
tions from J^e.m., by adding to the Lagrangian a counter term ^ c - m -SC 1 , i.e., 
writing eq. (8) as: 

(14) Js = ir -f- Jr 1 -f- =^ e.m. \ *^weak = 

= =^o i <>&i~t~ o ' • J ^ 1 4" V^e.m. o ' ' -^i) ~r =^weak • 

Therefore 8 e - m -^ ? 1 is determined by the following condition on the matrix 
element for the transition of one scalar meson into the vacuum extrapolated 
to zero four-momentum: 

(15) lim <0|(^ e .m. - S"*-*^, P> = . 



Weak interactions and the breaking of hadron symmetries 59 

In the following we will indicate by £30^ the whole isospin breaking term of 
S£ x -\- S^-J^, calling %e z a z it purely e.m. part: 

(16) £ 3 #3 = ^(1— Z)<?3 + Z £ 3^3- 

Any isospin violating amplitude can thus the splitted into two terms, i.e. : 

(17) (contribution of =2^ + ^ em -^ , 1 ) ++ (non-tadpole e.m. contributions) . 

The non-tadpole contributions correspond to graphs different from Fig. 2, 
e.g., of the kind shown in Fig. 3. 



■&y///ys?) 



Fig. 3. 



We will first discuss the two cases, where, due to the theorems of Bell 
and Sutherland [13], and of Dashen [14], the non-e.m. contribution to iso- 
spin breaking can be uniquely identified. 

These two effects are the violation of the Dashen sum rule for K + —K°, 
n + — n° mass difference, and the 7]-^37r decay. In these cases it is conve- 
nient to decompose as in eq. (16) the isospin breaking Lagrangian, and use 
the computational scheme: 

(18) (isospin breaking effect) = (contribution of e 3 (l — %)a^) + 

+ (total e.m. corrections) . 

To lowest order in jSf l9 and neglecting e.m. effects, the masses of pseudo- 
scalar mesons are: 

** H-^ + ^ + ^l- 



60 N. Cabibbo and L. Maiani 

+ *6l/ih 






where particle symbols stand for the corresponding mass squared, C is the 
reduced matrix element ||<jr|(r|7r>||, and 8M* n is the yj-tc mixing matrix element 
induced by a z . The Dashen sum rule implies that the combination 

(K + — K°) — (ti + — 7i°) = — 5260 (MeV) 2 

is not affected, up to terms of order e 2 SC U by the e.m. corrections. This 
allows a unique determination of the ratio e 3 (l— #)/e 8 according to: 

(19) ^ri = _ (^-^-(^-^ = 

V3£s 2(K— 7r) 

Using eq. (18) and the Bell-Sutherland results one concludes that the 
7) -» 3tt amplitude is given by 

(20) r=<37c| ea (l-z)*8h>. 

We note at this point that models of yj decay based on the a z interaction had 
already appeared in the literature. Only after Sutherland's result, however, 
it became clear that such an interaction could not be a consequence of elec- 
tromagnetism. 

Another class of models which have been advanced, is based on yj-u mixing, 
and describes the decay as y\^-k q ^3-k or y] -^rpm -^7r°7i7r. Bell and Suth- 
erland have shown that such models cannot work if e.m. interactions only 
are assumed. An insight into this result is given by the Dashen sum rule. 
Using £/-spin invariance one obtains: 

(21) SM; n =l{(r-^)-(^-^o)} 

which, combined with the Dashen sum rule, shows that the mixing cannot 
be ascribed to electromagnetism. 

In Appendix II we give a simple derivation of the 73 -> 3n amplitude 
based on eq. (20). The result of this calculation is: 



(22) r(yj -> 3tt) 



— y„ — -„, 2 

7T fc 8 



Fie, 



3f | £8 ^, Mjy u\-ui y y 



Weak interactions and the breaking of hadron symmetries 61 

where F n = 2MxgA/gxxn & 1.21 M„ is the pion decay constant; S is the7r+Tc _ 
invariant mass, Q = M ri —3m v . and y = (T n o—T n <>)/T n <>. Equation (22) gives 
a slope of the Dalitz plot distribution in excellent agreement with the experi- 
mental one. From the previous determination of e 3 (l — %)/e 8 , eq. (19), one 
obtains : 

(23) r(ti -> 7r+7i-7r ) sa 75 eV . 

Using the experimental branching ratio for the yj -> 7E + 7Tr7r° decay mode, 
one can transform eq. (23) into a prediction for the total Yj-width. One gets: 

(24) r^ & 330 eV . 

We do not try to attach an error to eq. (24). As an indication of the possible 
range of variation of r , we may mention that a more refined determination 
of our parameters e 3 , £ , e 8 , by G. Parisi and M. Testa [19], gives a values for 
e 3 (l — x)/ s s as high as 1.6xl0 -2 , which would lead to 

(25) r„ & 600 eV . 

Our prediction disagrees by a factor 3 -f- 6 from the value /^ =(2. 1 ±0.5) keV 
established [20] on the basis of the measurement of the yj -> yy width by the 
Primakoff effect [21]. We note, however, that eq. (24) is in agreement with the 
width obtained from 7i° lifetime and the use of SU 3 to relate the 7r°->2y 
and yj -» 2y rates [22]. A possible interpretation of the deviation of the 
experimental yj -> yy rate from the SU 3 value invokes the effect of yj-X 
mixing, with an abnormally large X°->yy coupling. We have not included 
in our calculation the effect of yj-X° mixing, but a rough order of magnitude 
evaluation indicates that this effect should not change our prediction by 
more than 10^-30 % (which is the general order of magnitude of SU 3 breaking). 
We cannot exclude that also in our case the discrepancy can be explained 
by some abnormally large SU 3 breaking effect connected with yj-X mixing. 
A critical reconsideration of the experimental determination of yj ^ yy 
might also be useful. 

We consider now the problem of the isospin breaking mass differences 
within the octet of stable, spin | baryons. 

Using our previous consideration, and in particular eq. (19), we may 
compute the effect of the nonelectromagnetic term e 3 (l— %)o 3 . We can then 
split the experimentally observed mass differences into an electromagnetic 
and a nonelectromagnetic part (see Table II). 

A first qualitative conclusion one can draw from Table II, is that the 
part ascribed to electromagnetism is either very small, or has the « natural » 



62 N. Cabibbo and L. Maiani 

Table II. 





M exp (MeV) 


«s(l — Z)^3 


e.m. 


£3 ff 3 


nontadpole 


p-n 


-1.3 


-2.0 


+ 0.7 


-1.6 


+ 0.3 


S"-S° 


4.9 


+ 3.0 


+ 1.9 


2.4 


+ 2.5 


S°-S+ 


3.1 


+ 3.0 


+ 0.1 


2.4 


+ 0.7 


H -E° 


6.6 


+ 4.0 


+ 2.6 


3.2 


+ 3.4 



sign, corresponding to charged particles heavier than neutrals. The analysis 
of the baryon mass differences can also be carried out along the first of the 
procedures indicated above, eq. (17). This corresponds to the well-known 
tadpole analysis of Coleman, Glashow and Socolow [23]. Their results appear 
in the last two columns of Table II. Comparing the second and fourth columns, 
one can obtain the value of (1— #), i.e. the ratio of the nonelectromagnetic 
tadpole to the total tadpole contribution. The result is: 

(26) 1-*«1.2. 

Again, we are not able to attach errors to this determination, which however 
should not exceed rm 20%. The parameter %, which determines the strength 
of the e.m. tadpole is then expected to be: 

(27) Z^-0.2. 

It emerges from Table II that actually the non-e.m. terms provide the main 

part of the isospin breaking mass differences. This suggests the possibility 

that one can obtain rough estimates of other Al = 1 isospin breaking effects 

by considering the contribution of this interaction. 

As an example let us consider the problem of corrections to tt-JV scattering 

lengths. 

We find that the charge independent relation: 

(28) a(r-p -> 7r°n) = — = {a(n+p) - a(^"p)} 

V z 

after inclusion of the £ 3 (1 — %)a z term is modified into: 

(29) a(7r-p -► 7i0 n ) = i±- (a(u+p) - a(n-p)} , 

V l 



Weak interactions and the breaking of hadron symmetries 63 



where: 



(30) d = (M ° ^p)°on-e.m. ^ QJ x 1Q _ 2 _ 

This correction is about one order of magnitude smaller than the present 
uncertainties on the measurements of the amplitudes involved, so that a 
test of eq. (29) is at present impossible. 

5. - We have anticipated in Sect. 3 that weak interactions may lead to correc- 
tions to hadron processes competitive with strong and e.m. effects. The 
study of this poses a very complicated mathematical problem, which has 
been solved up to now only in part and for expecially simple cases. 

Let us assume weak interactions to be mediated by a vector boson coupled 
to the SU z 0SU 3 hadron currents. The simplest model consistent with our 
present understanding requires only one charged boson, coupled to the current 
given in eq. (3). 

The possibility of large corrections due to weak interactions is indicated 
by the appearance of highly divergent integrals in the computation of these 
effects with perturbation theory. If one introduces a regularization procedure 
through a cut-off A y one finds at order 2« divergent terms of order (GA 2 ) n , 
where G is the Fermi constant, followed by less divergent terms of order 
GiGA 2 )™' 1 , G(GA 2 ) n ~ x \ogA, etc. 

A way to extract information from the perturbation theory under such 
circumstances has been proposed by T. D. Lee [24]. It consists in resumming 
the series according to the order of divergence, and then letting A -> oo. 
This procedure would lead to the following expression, for the regularized 
^-matrix : 

(31) S = S^\GA 2 ) + GS^\GA 2 ) + ... , 

where S i0) (GA 2 ) contains all the terms of order (GA 2 ) n , i.e., the leading divergent 
terms, S^iGA 2 ) contains the terms of order G(G'/l 2 ) w ~ 1 and so on. Going 
to the limit A~>oo, if S i0) , S^\ etc. have finite limits, one would have ob- 
tained a new expansion of S in powers of G with finite coefficients. In par- 
ticular the first term S {0) (GA 2 ) gives the S-matrix a contribution completely 
independent from the Fermi constant. 

This program is complicated by the presence of logarithmic divergences, 
which would probably require a separate treatment, and by the fact that 
only the first term of eq. (31) is in general unambiguous. The subsequent 
terms are expected to depend upon the way in which the theory is regularized. 
We will restrict our analysis to the first term in eq. (31), which is expected 
to contain all the possible « strong » effects due to weak interactions, and 



64 N. Cabibbo and L. Maiani 

leave open the important problem of the other terms, which are expected to 
give the true, order G or higher, weak corrections. 

Let us assume strong interactions to be described by a Lagrangian: 

\j1) Jz s = Jzq -\- ~z? i , 

where J? is invariant under SU 3 (x)SU 3 , and <£ x represents the symmetry 
breaking. Hadronic weak interactions are assumed to be described by the 
Lagrangian: 

(32) ^weak = S(^+h.C.), 

where /„ is the current eq. (3), and g is related to the Fermi constant by: 
G = v / lg 2 M~ 2 . Let us define: 



(33) 



e+=Jd 3 x/ (x) 



It is instructive to consider a second order calculation of weak corrections to 
any strong amplitude A(oc -> (5), where a and (5 are hadronic states. This is 
given by: 

(34) ^ = -£J d .,_^(_,„ + 4*). 

r exp [iqx] [<fi\T(Jp{x)J*(0))\K> + c.c] d*x . 



■/• 



Assuming the validity of Bjorken's limit, quadratically divergent terms arise 
only from the q /i q v term, and are therefore connected with the nonconserva- 
tion of the weak current, i.e., to the symmetry breaking term =2^. Introducing 
a cut-off A, by standard manipulations one finds the quadratically divergent 
term to be [25]: 

(35) 8< 2 U=-/</ff|8<2)js? 1 | a > 

with 

GA 2 

(36) S^ = 2"{[S + , lQ ~> ^ + [e "' [Q+ > ^D) ' 

From eq. (36) we see that §( 2) A is equivalent to the shift in A caused by 
the addition of a piece S^jSfj. to J^, when S^J^ is treated perturbatively 
to lowest order. If one goes to the fourth order weak corrections, one finds 



Weak interactions and the breaking of hadron symmetries 65 

a term which corresponds to the second order perturbation in 8^Sf lt as 
well as other terms which should be identified with a further shift, 8( 4) J^ l5 
of J?!, treated to lowest order. 

This circumstance suggests that the leading weak corrections to any 
strong process might be equivalent to a modification of S£ x \ 

(37) JS? 1 -J-JSf 1 + 8 weak JS?i, 
where : 

(38) 8 weak ^ = W>& x + S (4) ^i + ... . 

In the general scheme of weak interactions we have described till now, this 
is only a conjecture as yet unproven, and we have not even a complete analysis 
of SWjgf 7 !. There are however models where this conjecture can be proven, 
and the structure of S weak J^! written down in detail. Two models of this 
kind were studied in ref. [1]. They are the free quarks model and models 
allowing nontrivial strong interactions, but where weak interactions are trans- 
mitted by a neutral vector boson. The neutral boson is coupled to the cur- 
rent J^, defined by the relation: 

(39) Q=jd*xJ (x) = i[Q + , QT}. 

The second model was analyzed under the assumption that the breaking 
term & x transforms as a (3, 3) © (3, 3) (which is necessarily the case for 
quarks). In both cases it was found that, if £P X is characterized by a 3 x 3 
matrix h (see Appendix I, eq. (1.1) and (1.3)), then & x + 8 weak ^ corresponds 
to the matrix Ti: 

(40) h = h + 8 weak /* = h — g {A+, X~) h , 

where X+ is the matrix defined in eq. (6), X~ its Hermitian conjugate, and | 
is a real parameter. This result is the same that one would obtain at second 
order, eq. (36), if one put £ = GA 2 . In the quark model g remains as a 
parameter as yet undetermined, whereas in the neutral case £= 1. 

An important feature of this result is the following. Suppose we start 
from a parity and strangeness conserving ^ x , i.e. with a diagonal and real 
matrix h. Equation (40) then implies that h is also a real matrix so that, by 
Theorem 1 of Appendix I, h is equivalent to a real and diagonal matrix h D : 

(41) h=Wh D V. 

This means that the leading weak corrections do not cause a breaking of 
parity and strangeness, which should only arise at order G. This however 



66 N. Cabibbo and L. Maiani 

is a peculiar property of the (3, 3) © (3, 3) behavior of the breaking S£ x . 
In fact one can easily show that if ££ x contains a part which transforms as 
(8,1)0(1,8), this result is not valid, and strong parity violations arise. 
Proof of this is given in Appendix III, both using the second order calcula- 
tion, eq. (36), and the neutral vector boson model, treated to all orders. 

We have emphasized in Sect. 1 the remarkable connection between weak 
currents and strong interaction symmetries. The result we have just quoted 
together with the indication discussed in Sect. 2 that symmetry breaking is 
of a (3, 3) © (3, 3) kind, adds a new piece of evidence for the strict dynamical 
intertwining of weak and strong interactions. Coming back to the case 
when J£\ belongs to a (3, 3) © (3, 3) we note two facts. 

First is that the transformation (41), which reinstates parity and strange- 
ness, changes the value of the weak interaction angle. We will call 6 the 
uncorrected angle (i.e. the angle in the frame where h is diagonal) and 6 the 
one in the frame of h D , i.e. the physically observed one. The angle appearing 
in eq. (40) through X + and X", is obviously 6 , and 6 is a function of |, h, and O , 
as discussed in Sect. 5 of ref. [1]. 

We finally note that even if h conserves isospin symmetry, h D will in 
general not do so. Weak corrections thus seem to give a natural explanation 
for the origin of the non-e.m. isospin breaking which is required by the 
phenomenological analysis of Sects. 2 and 4, and which is not easily under- 
stood on the basis of strong interactions dynamics. 

6. - We have studied till now the separate effects of e.m. and weak cor- 
rections on hadron dynamics. In this Section we will collect these different 
results, and introduce a self-consistency condition [1] which links at a dynam- 
ical level, the different interactions and allows a determination of the angle d 
in terms of other parameters. 

Starting from the equation: 

(42) se = se* + sei + ^ e . m . + ^ 

and following the results of Sects. 3 and 5, we separate the leading weak 
corrections and e.m. tadpole contributions according to: 

(43) ££ = Cf Q + ^ + §weak ^ + §e.m. ^ + 

+ (^e. m .— 8 e - m -=^l) + (&— 8 Weak ^i) • 

The last two terms correspond to nontadpole e.m. corrections and to non- 
leading, i.e. order G, weak corrections. As explained in ref. [1], Sect. 3, 
ge.m.^ depends in a peculiar way on the explicit breaking. 



Weak interactions and the breaking of hadron symmetries 61 

If we define: 

(44) 8 em - jSf x = I Tr (M + SA e - m - + $h e - m -* M) 

the result of ref. [1] is that S/* e - m - is diagonal in the same SU 3 (g)SU 3 frame 
as the explicit breaking. The strength of SA e - m - is, on the other hand, mainly 
determined by the dynamical breaking of SU 3 ® SU 3 . Since this, according 
to the discussion of Sect. 2, reduces SU 3 (x) SU 3 into a nearly exact SU 3 , 
we expect 8/* e - m - to be a nearly exact £/-spin singlet. 

We have seen that the effect of weak corrections is to change the explicit 
breaking from ^(/z) to & x (h D ). Then U e - m - will still be diagonal with h D 
and will be changed only slightly by this modification, as it is expected to 
depend mainly upon the dynamical breaking. The combined effect of weak 
and tadpole e.m. interactions then changes the explicit breaking Lagrangian 
^(h) appearing in eq. (42) into Sejfrv + 8/* e - m -). 

The self-consistency condition of ref. [1] requires h to be stable under 
these effects, i.e.: 

(45) h D + §/i e - m - = h . 

Equation (45) gives a relation among the parameters a, /?, y, g, and either 6 
or O . We will not report here the complete structure of eq. (45), but simply 
give the results, valid in the case where |/S/y| < 1 and |£j8/y| < 1. One ob- 
tains the relations: 

(46) «-.^ W 



(47) I 



y Vi-l/2 

a — jff a e.m._ j ge.m. 



Equation (47) implies that only a portion I of the isospin breaking contained 
in h D + 8/z e - m - is due to pure electromagnetism. The remaining fraction 
(1_|) i s nonelectromagnetic, so that £ has to be identified with the param- 
eter % introduced phenomenologically in Sect. 4. Substituting into eq. (46) 
the result found there, eq. (27) and the value for 0/y deduced from eqs. (12), 
(22) and (26), one finds a prediction for 6: 

(48) 6 ** 0.25 . 

7. - An outlook. 

We have presented here a review of some recent developments in the 
study of the dynamical interplay of weak, electromagnetic and strong in- 
teractions. 



68 N. Cabibbo and L. Maiani 

The best established consequence of this approach is the natural expla- 
nation for a new kind of isospin breaking, not uniquely electromagnetic. 
This new term is required for the interpretation of experimental data con- 
cerning 7^-decay and the mass differences of pseudoscalar mesons. 

A second result which might suggest ideas of future development is the 
realization of the special dynamical role of a symmetry breaking of the 
(3, 3) © (3, 3) kind, which gives a fair description of experimental facts. 
Among « simple » breaking schemes this is the only one which allows parity 
to be conserved by the leading weak corrections. 

Finally, the introduction of a new hypothesis, of a self-consistency con- 
dition among weak, electromagnetic, and strong interactions gives a rela- 
tion between 6 and other phenomenological parameters which yields an 
excellent prediction for 6. 

Although these problems have been partly clarified, there remains a great 
amount of work to be done on them, in particular in the treatment of higher 
order corrections with a W ± boson. The model with a W boson, although 
suggestive, is far from representing the real situation. 

Other serious problems relating to higher order weak corrections have 
not be touched. The most serious one is that of the selection rules observed 
in weak decays (A5'<2, A/<|, no neutral currents, etc.). Simple compu- 
tations at second order indicate a breakdown of each of these rules at order 
G X GA 2 z& G. In order to agree with experiment, the theory should com- 
pensate this breakdown, perhaps at higher orders. 

On the other hand the solution of the parity problem at the GA 2 level 
leads some credibility to the hope that these harder problems will also be 
solved. 



Appendix I. 

In this Appendix we give a simple algebraic proof of the Michel-Radicati 
result. 

The possible dependence of G on e 3 , s 8 , s is restricted by symmetry con- 
siderations. To exploit them it is convenient to give a more general definition 
of the symmetry breaking Lagrangian. 

Starting with the densities at and m defined in Sect. 2, we define a 
3x3 matrix M: 

8 

(1.1) M = 2>, + *rc<)A<, 

where h are the usual Gell-Mann matrices (A = V2/3). Under an element 



Weak interactions and the breaking of hadron symmetries 69 

(U, V) of SU 3 ®SU 3 , M transforms as: 

(1.2) M->UMV i U, V= unitary, uninodular 3x3 matrices. 

Since J^ is an element of this (3, 3) © (3, 3) representation of SU 3 ® SU 3 
it can be identified by a 3x3 matrix h according to: 

(1.3) J2\(A) = J Tr (A + M + M^h) . 
Under a transformation (U, V), £P X transforms as: 

(1.4) &&)->&&') 

(1.5) h'=U*hV 

Definition. Any pair of matrices h and h', obeying eq. (1.5) are called 
equivalent: /z~/z'. In fact =2\(/z) and j£?i(A') have the same physical content, 
because they are related by a change of basis under which ££ Q is invariant. 

Theorem 1. If det/* is real, h^hj) where: 

/a 0\ 

(1.6) h D = I /? 1 «>/?,/ being real numbers . 




A proof of this theorem is given in ref. [1]. In this language, the bootstrap 
condition which determines the breaking must have the form: 

(1.7) $G(h) = . 

Since two equivalent matrices have the same physical content, one must 
have: 

(1.8) G(h') = G(h) if h'~h. 

Then G(h) is a function of the three SU 3 (x) SU 3 invariants one can con- 
struct out of h. These are: 

(1.9) x 1 = T 2 (hh i ), x 2 = Tr [(M + ) 2 ] , x 3 = deth. 
For diagonal matrices hp, we have: 

(1.10) ^ = a 2 + i8 2 +y 2 ; jc 2 = a* + /S 4 + y* ; x 3 = ofty. 

In the following we restrict our study to solutions which are parity conserving, 
i.e. such that det/z is real. 



70 N. Cabibbo and L. Maiani 

Theorem 2. Up to an equivalence, all the parity conserving solutions to 
eq. (1.7) are obtained as solutions of the equation: 

(1. 1 1) %G(x x ((x, ft, y), x 2 (cc, ft, y), x 3 (oc, ft, y)) = , 

the variation being done in respect to a, fi, y, assumed to be real numbers. 
This means that we can restrict in eq. (1.7) to real diagonal matrices h D , thus 
recovering the variational principle given in Sect. 3. 

Proof. For any h we have: 

h = u*h D V 

with suitable unitary, unimodular matrices U, V, and a diagonal h D . Moreover : 

h + 8h = U'\h B + Bh D )V' 

where U' and V are infinitesimally different from U and V and §h D is diagonal. 
One then has, using eq. (1.8) 

8G = G(h + 8/0— GQi) = G[U'\h D + M D )V']—G{U^h D V) = 

= G{h D + %h D )-G{h D ). 

Although a, ft and y can vary through the whole 3-dimensional space, x 1 , 
x 2 , x 3 are restricted to a definite domain D. The boundary of D is composed 
by continuous surfaces, joined by edges which meet at singular points. The 
equations for the different elements of the boundary can be obtained as follows : 
Define vectors H = (oc,fi,y) and X = (x 1? x 2 , x 3 ). To a variation $H there 
corresponds a variation 

The elements of the boundary are characterized by special properties of d E X. 

I'l. Characterization of surfaces. - Denote by n(x) the normal to the 
surface pointing outwards from D. The variation 8 S X must be orthogonal 
to n for any choice of 8H: 

(1.13) n-S H X=0. 

Proof. For any $H, X must always remain in D, i.e. n-S s X<0. Since 
$ H X is linear in 8H, inequality can only be fulfilled as an equality. 

1*2. Characterization of the edges. - Denoting by l(x) the unit vector 
tangent to the edge, by the same argument as before one must have: 

(1.14) $ B X parallel to /. 



Weak interactions and the breaking of hadron symmetries 71 

I'3. Characterization of singular points. - Arguing as before, one gets: 

(1.15) S H X=0. 

Conditions 1, 2, and 3 are equivalent to the requirements that the Jacobian 
matrix : 

/2a 2/3 ly 

\Py yoc ocfi, 

has a rank equal to two, one, and zero, respectively. One finds that, on the 
boundary, one must have: 

(1.17) detJ = 8(a 2 -W 2 -y 2 )(y 2 -a 2 ) = 0. 

This implies that on the boundary of D, two eigenvalues of h D are equal 
(the solution, e.g., <x, — a, y is equivalent to the solution a, a, y). The sur- 
faces then correspond to SU^®U X . On the edges, the Jacobian matrix 
should be of rank one. This implies either: 

i) a 2 = /3 2 = y 2 , i.e. exact SU S . These equalities have two inequivalent 
realization: (a, a, a) and (—a, —a, —a), corresponding to the two edges: 



x a 1 

(1.18) x 2 =-^; x 3 =±-xl; 



r 2 
J: 

3 ' ~ 3 ^27 

or 



ii) a = ft = 0, y^O and permutations, i.e. exact ,SC/ 2 (x) »St/ 2 . Thus ii) 
corresponds to a single edge, where one has: 

(1.19) x 2 = x\; x 3 = 0. 

There is only one singular point, where /= 0, i.e. the origin corresponding 
to exact SU 3 (g) SU 3 . Solving eq. (1.17) for a, and substituting into eq. (1. 10), 
one gets a parametric representation of the boundary. In Fig. 4 we report 
a section x x = const of D. Points A, B, C are the intersection of the edges 
with this plane. Points B and C correspond to eq. (1.18) and A to eq. (1.19). 
The following theorems establish the conditions for the existence of extremal 
points of G on the boundary of D: 

Theorem 3. If X is a point on the surfaces composing the boundary 
of D, and is an extremal point with respect to the values assumed by G on 
this surface, then X is an extremal point for G in D. 

Proof. The condition for X to be an extremal point of G in D is: 

VG-8 H *=0. 



72 



N. Cabibbo and L. Maiani 



If X belongs to a boundary surface, § H Xis tangent to it, by eq. (1.13), and 
the components of VG along the surface are zero by hypothesis. 




-0.2 



-0.1 



o 
Fig. 4. 



0.1 



*3 



0.2 



Analogously one can prove the following theorems: 

Theorem 4. If A" is a point on an edge of D, and is an extremal point 
with respect to the values assumed by G on this edge, then X is an extremal 
point for G in D. 

Theorem 5. If X is a singular point on the boundary of D, it is an ex- 
tremal point for G in D. 

In spite of these theorems, we are still not in position to prove the 
Michel-Radicati result, apart from the existence of an extremal point cor- 
responding to the case of no breaking. The reason is that, since the boundary 
of D has an infinite extension, there is no guarantee for the existence of an 
extremal point on it. In order to conclude the proof of the theorem, a further 
assumption is needed, which permits the search of an extremal point in a 
region of finite extension. 

We assume that the bootstrap conditions are not able to fix the scale of 
the masses, i.e. the scale of a, /S, and y. This implies that G depends only 
upon the ratios x 2 /x|, x z \x\ and is therefore constant along the edges of the 
boundary. Each point of this edge is therefore (by Theorem 4) an extremal 
point of G on D. Moreover one can limit the search of extremal points of G 
on the cross-section x 1 =\, analogous to that given in Fig. 2. If G is limited, 
it will have at least one point of minimum and one of maximum on this 
(finite) domain, which may or may not coincide with the extremal points A y 
B and C. 



Weak interactions and the breaking of hadron symmetries 73 



Appendix II. 

We give here a derivation of the t\ -> 3iz amplitude given in Sect. 4 
eq. (22), using the method of nonlinear realization of chiral symmetry given 
by Coleman, et al. [26]. These authors have shown that the results of current 
algebra and PCAC can be obtained by the use of phenomenological Lagran- 
gians in the tree graphs approximation. They have also given rules for writing 
down these Lagrangians, and have shown the uniqueness of such a description. 

The eight pseudoscalar mesons are represented in this formalism by a 
3x3 traceless matrix: 

(11.1) Jt=2V< (i=l,... ,8). 

i 

The SU 3 (x) SU 3 symmetric Lagrangian for amplitudes involving pseudoscalar 
mesons only is: 

(11.2) ^o^-yTrOv^), 

where p M is the covariant derivative of the pseudoscalar fields, defined in 
ref. [26]. An expansion of p^ in powers of n is: 

(II 3) p,, = - — dpTi — -= - 3 [[3^, n\ n\ - ... . 

The phenomenological form of the symmetry breaking ^ is: 
(IL4) JSfi = e,(l - x) S 3 + e 8 S 8 + e S , 

where St are functions of jr-fields which transform according to the (3, 3) ©(3, 3) 
representation of SU 3 ® SU 3 . The choice of these functions is also unique, 
and they can be obtained from the formal definition: 



(115) S + iy 5 P = exp 






A development of the right-hand side of eq. (II. 5) in power series gives: 



(116) S=l—= i 7i i + T = i ^- 



1 2 , J_ 

F 2 * + 3F*~ 



01.7) P=^7l-~7l*+.... 

Thus jSf-t contains terms bilinear in the jz-fields. The coefficients of these terms 
give the mass matrix, including 7)-7r° mixing. The elements of the mass matrix 



74 



N. Cabibbo and L. Maiani 



agree with those given in eq. (18) of Sect. 4, provided one sets: C = 4F^ 2 . 
We can diagonalize this matrix, the lowest order in e 3 (l — #)/e 8 introducing 
two new fields fc and r\ which describe the physical r] Q and n Q fields: 



(II.8) 



71 = 71^ — fj s , 



V =r ls- 



2e 8 
2e« 



The amplitude for vj -> tttt+tt - can then be calculated by the terms of J§? + jS^ 
quadrilinear in n. There are in principle three such contributions. Two of 
them come, through the action of mixing, from if and from s s S 8 + s S . 
Actually the second contribution vanishes in this case. The last contribution 
comes directly from the term £ 3 (1 — x)^s- The two nonvanishing contribu- 
tions are: 



(II.9a) 
(11.96) 



2 %(!-*) 

3 n* 



(3S—P 2 —q 2 — k 2 —p 2 ) 



3 e a F 2 K * n) ' 



~8*n 



where P, q, k, p are the momenta of vj, tc+, tt , 7i, and s = (q + k) 2 = (P — p) 2 . 
The total amplitude is then: 



(11.10) T 



2* 8 (1-Z) 



Kh 



'■r^-\{P*-M$-\{q* + k*+p*-3m*^ 



which reduces on the mass shell to eq. (22) given in Sect. 4. From the well- 
known relation: 



(11.11) 



d»A)=-i[Q\,^} 



where Q\ are the axial charges, and through the definitions, eqs. (II.4) and (II. 5), 
we see that the divergences of axial currents turn out to be proportional to 
the Pis. This means in turn that PC AC is valid only up to terms trilinear 
(or higher) in jr-fields. 

Even in presence of the £ 3 (1— %) breaking term, one can define suitable 
combinations of A* and A*, whose divergences are proportional to the n 
or rj fields, eq. (II. 8), up to terms of higher order in pseudoscalar fields. In fact 
one has: 



(11.12) 



A* 



A\ 



A* \ _L Al 



C 3 A3 



Weak interactions and the breaking of hadron symmetries 75 

and 



(11.13) 



F m 2 „ 
F M % 



This means that in processes involving less than three pseudoscalar mesons, 
and in the three graphs approximation, the 7r-fields can be treated as diver- 
gences of suitable currents. In particular they will satisfy the Adler consistency 
condition, and moreover the effect of the S^-breaking reduces in this case 
to the effect of current mixing, eq. (11.12). 

For processes with many pseudoscalar mesons the additional terms, 
present in axial current divergences in this scheme, cause a failure of the Adler 
consistency condition. Choosing a nonlinear realization different from the 
« standard one » given in ref. [26], one could restore the Adler zero's for 
pions. This however does not affect on the mass shell amplitudes since, as 
shown in ref. [26], these amplitudes are independent from the particular 
representation chosen. 

We consider now n-Jf scattering in the soft pion limit. One can treat 
this according to the method of Weinberg, using as interpolating fields the 
axial divergences: 

V^ a A ± 

for iz+ and -nr, and eq. (II.8) for ic°. One then easily finds that there is no 
modification to the Weinberg formula for A(iz + p) — A(iz-p), whereas the 
charge exchange amplitude is given at threshold by: 

, TT1 „ . A + — A~ ( (Mp-M n )non.e.m. \ 
(11.14) ^^-—.^l _ j, 

where (M p — M n )non-e.m. indicates the non-e.m. proton-neutron mass differ- 
ence, as given in Table II, of Sect. 4. 



Appendix III. 

In this Appendix we consider the leading weak corrections to hadron 
processes in a fictitious model where nonleptonic weak interactions are ascribed 
to the coupling of a neutral vector boson W only. We assume strong in- 
teractions to be described by the Lagrangian 

(HI. 1) ^ s = ^ (v»<, 3„y *) + ^i(v') . 

where ^ is invariant under SU 3 (g) SU 3 and J? lt the breaking term, is as- 
sumed not to contain derivatives. \p l represent a set of hadron fields, having 



16 N. Cabibbo and L. Maiani 

definite transformation properties under SU 3 (x) SU 3 . If we call F* the sixteen 
generators of SU 3 (x) SU 3 , we have: 

(III.2) [F°,y> i ] = iT? j y>1, 

where the matrices T% constitute a representation of the generators F a . 

Weak interactions with the neutral vector boson are assumed to arise 
from the minimal substitution: 

(ni.3) a^'-^v' + fc^^, 

where Tis the representative of the current to which Wis coupled. To connect 
this model to the realistic one discussed in Sect. 5, where weak interactions 
are mediated by charged vector bosons, we choose T to correspond to the 
third component of weak isospin, i.e. T corresponds to the charge Q 3 
defined as: 

(H.4) Q s = m + ,Q~] 

with Q ± defined as in eq. (33) of Sect. 5. In the quark language: 
e 8 = V^A»(l+y 8 )v' 



(HI 5) 



a 3 =i[A + ,n 





sin 2 



6 is the weak interaction angle in the SU 3 ® SU 3 basis in which J^ is parity 
and strangeness preserving. We also introduce minimal electromagnetic 
coupling, and write: 

(111.6) se = J2? (y', D^f) + ^(y»*) + J^ ph + ^ w , 

=^ph and i?w are the free photon and W-meson Lagrangians, and we have 
defined D^ip* to be: 

(111.7) Drf = dptp* + igWJ i0 1 + ieA&yipt , 

Qtj being the matrix representing the electric charge. 

To isolate the leading weak corrections, it is convenient to make use of 
the Stuckelberg formalism [27], following the work of ref. [24]. This con- 
sists in decomposing W ^ into two terms: 

(IH.8) w^w^-^-dJ. 



Weak interactions and the breaking of hadron symmetries 11 

With a suitable modification of J^ w , W^ has the propagator: 

/TT CVv o (tv 



\1±.7J 








<7 2 -^w' 


whereas the field 6 


is 


given 


the 


propagator 


(III. 10) 








1 


d*-Ml' 



The coupling of W^ gives rise by itself to a renormalizable theory, provided 
the \p l have spin \ or 0, similar to electrodynamics. All leading divergences, 
coming from the q^q v term in the W^ propagator, are now associated with 
a derivative 6 coupling. We perform now the canonical transformation: 



s™ 






(III. 11) ^* = jexp 

One easily finds: 

(HI. 12) D„yt* = U tJ D li yt, 

where 

(111.13) D„r = drf + HgW^Ty + eA„Qu\ & , 

J? being invariant under chiral transformations, we have: 

(HI. 14) JS^l/y, UB, W) = J? (V, D, y) . 

We have thus eliminated from J? the dangerous field 6. On the other hand ^ 
is not chiral invariant, so that: 

(111.15) & x {U{x)$) = expLz J-g^lj^exp i-JL&Oix)] . 

As explained in ref. [24], the leading weak corrections to hadron processes 
are obtained considering all the graphs where a 0-line starts and ends at the 
same vertex. This amounts to calculate the expectation value of ^' 1 {U{x) ! ^) 
in the vacuum state for 6-mesons, thus defining: 

(III. 16) Se x + S weak ^i = <^i(U(x) #)> . 

To go further, we must specify the transformation properties of SC X . The 
case where l£ x transforms as a (3, 3)©(3, 3) has been discussed in detail in 
ref. [1], and yields the result quoted in Sect. 5, eq. (40). With analogous 
techniques, one could treat the other possible forms for J^. 



78 N. Cabibbo and L. Maiani 

We are here interested in the problem whether eq. (III. 15) can give rise 
to a new breaking J^ + S weak j2\ equivalent to a parity conserving one, as 
was the case for the (3, 3 (3, 3). 

Let us consider the case (1, 8)© (8, 1). A basis for such a representa- 
tion is given by a set of eight scalar and eight pseudoscalar densities a\{x) 
and d\{x). We collect these densities into two, 3x3 matrices, according to: 

8 

D = J t d t (x)X t 

The combinations: 

(III. 17) D± = D±D 5 

transform as the (1, 8) and the (8, 1) representations, respectively. A parity 
conserving J^, would have the form: 

(III. 18) J2\ = 2 Tr (DH) = Tx{D + H + D~ H) , 

H being a numerical, 3x3 traceless Hermitian matrix. Because of the 
1 + y 5 structure of the current, only the D + component of <£ x is modified 
by weak corrections, and eq. (III. 15) reduces to: 

(111.19) ^ 1 (U(x)D)=Tr(D+exp\i 1 ^^d(x) Hexp\-i^tfd(x) +D~HJ. 

This means that the effect of the 0-field is to impart the left-handed part of ^ 
an SU 3 transformation depending upon the quantized density B(x). According 
to eq. (III. 16), the modified breaking operator is to be obtained by averaging 
eq. (III. 19) over the value of 6(x) in the state with no particles. One then 
finds : 

(111.20) Se x + S weak Se x = Tr (D+ H' + D~H). 

In order for parity to be conserved, it should be possible to reduce H' to H 
by an SU Z transformation. A necessary condition for this is: 

(111.21) Tr {H'H 1 ^) = Tr (Hrf) . 
However, one easily sees that: 

(111.22) Tr({U(x)HU\x)y (U(x)H'U\x)y )<TriUHU'UH i U} ^Tr(HH f ) 

equality being attained if and only if 

(111.23) [X\H] = 0. 



Weak interactions and the breaking of hadron symmetries 79 

If A 3 is given by eq. (III.5), eq. (111.23) can be satisfied either if d = 
(and 6 = 0), or if H is proportional to the electric charge matrix Q (where 
H'=H). 

The analysis we have formally carried out in the neutral boson model 
can easily be applied to the second order computation eq. (36) of Sect. 5. 
One obtains again the result eq. (111.20) where: 

H'= H- GA 2 {[X+, [r, HJ] + [r, [A+, H]]} . 

Up to terms of order (GA 2 ) 2 , one has: 

(111.24) Tr (H'H") - Tr {Hit) = 

= - GA 2 {Tr [H*([?i + , [r, H] + [r, [A+, H})] + h.c.} = 
= - 2GA 2 {Tr ([H \ X + ] [A", #]) + Tr ([# + , r ] [A + #])} = 
= -2G/l 2 {Tr([r, i/]) + [r, #] + Tr([A+, /f]) + [r, H]}<0. 

We note that if d ^0 the condition [A± i/] = 0, necessary to have an 
equality in eq. (111.24), cannot be realized by a diagonal traceless matrix. 
This proves that if Se x has a component along (8,_1)© (1, 8), parity is 
violated at the order GA 2 , contrary to the case of (3, 3) © (3, 3). 



REFERENCES 

[1] N. Cabibbo and L. Maiani: On the origin of the weak interaction angle II, in Phys. 
Rev. (to be published). 

[2] M. Geix-Mann, R. J. Oakes and B. Renner: Phys. Rev., 175, 2195 (1968). 

[3] S. Glashow and S. Weinberg: Phys. Rev. Lett., 20, 224 (1968). 

[4] L. Michel and L. Radicati: Breaking of the SU 3 xSU 3 symmetry in hadron physics, 
preprint (IHES and Pisa). 

[5] R. P. Feynman and M. Gell-Mann: Phys. Rev., 109, 193 (1958). 

[6] Y. Nambu and G. Jona-Lasinio: Phys. Rev., 122, 345 (1961); 124, 246 (1961). Many 
effects due to spontaneously broken symmetries (ferromagnetism, superconductivity, 
superfluidity) were already known from many-body physics, but it was still not real- 
ized that these were different manifestations of the same general phenomenon, and 
the crucial role of massless bosons was not entirely understood. 

[7] S. Weinberg: Review talk, Intern. Conf. High-Energy Physics (Vienna, 1968). 

[8] M. Gell-Mann and M. Levy: Nuovo Cimento, 16, 705 (1960). 

[9] N. Cabibbo: Phys. Rev. Lett., 10, 531 (1963). 
[10] For a recent assessment, see H. Filthuth: Proc. Topical Conf. Weak Interactions, 

CERN (Geneva, 1969). 
[11] The algebraic formulation of universality for the vector part of the current (N. Ca- 
bibbo: Reports of Erice School 1963) consists in requiring that the associated charge 
is a well-normalized raising operator. A complete formulation, involving also axial 
current, has been given by M. Gell-Mann: Physics, 1 (1965). 



80 N. Cabibbo and L. Maiani 

[12] M. Gell-Mann: Phys. Rev., 125, 1067 (1962). 

[13] D. G. Sutherland: Nucl. Phys., B2, 433 (1967) which contains references to pre- 
vious work. 

[14] R. Dashen: to be published. 

[15] S. Coleman and S. Glashow: Phys. Rev. Lett., 6, 423 (1961). 

[16] R. E. Cutkosky: Particle symmetries, Reports of the 1965 Brandeis University Sum- 
mer Institute in Theoretical Physics (New York, 1966). 

[17] L. Michel and L. Radicati: Proc. Fifth Coral Gables Conf. Symmetry Principles at 
High-Energy (A. Pearlmutter, C. A. Hurst and B. Krsunoglu eds.) (New York, 1968). 

[18] N. Cabibbo: Hadrons and their interactions (A. Zichichi, ed.). Reports of the 1967 
Intern. School of Physics «Ettore Majorana», Erice (New York, 1968). 

[19] G. Parisi and M. Testa: Nuovo Cimento (to be published). 

[20] G. Salvini: Riv. Nuovo Cimento, 1, 57 (1968). 

[21] L. Bemporad, P. L. Braccini, L. Foa, K. Lubelsmeyer and D. Scmitz: Phys. Lett., 
25 B, 380 (1967). 

[22] N. Cabibbo and R. Gatto: Nuovo Cimento, 21, 872 (1962). 

[23] S. Coleman and S. L. Glashow: Phys. Rev., 134 B, 671 (1964); R. Socolow: Phys. 
Rev., 148 B, 1221 (1965). 

[24] T. D. Lee: CERN preprint 68/940/5 - TH 940 (1968). 

[25] See, e.g., R. Gatto, G. Sartori and M. Tonin: Phys. Lett., 28 B, 128 (1968). 

[26] S. Coleman, J. Wess and B. Zumino: Phys. Rev., 177, 2237 (1969); C. Callan, S. Co- 
leman, J. Wess and B. Zumino: Phys. Rev., 177, 2247 (1969). 

[27] E. C. G. Stuckelberg: Helv. Phys. Acta, 11, 225 and 299 (1938). 



The Role of Complexity in Nature. 



G. Cocconi 

CERN - Geneva 



1. - A common aim of people interested in science is that of improving 
the comprehension of the phenomena that can be observed in the world. 
It happens that the variety of the panorama that falls under our senses is 
so great that the present degree of understanding is much diversified. 

In attempting a classification, we tend to divide the various branches of 
science into two groups: the « exact » and the « nonexact », a subdivision 
that reflects the degree by which the language of mathematical analysis is 
used in the interpretation. Typical of the exact sciences are astronomy, 
physics, etc., while the nonexact are to a great extent the biological sciences, 
the more so the more they deal with the behaviour of living organisms. 

This separation, however, is not merely methodological. It also reflects 
a more fundamental difference in the nature of the phenomena studied 
by the disciplines that are grouped under the two labels. The exact sciences 
have to deal with phenomena that, at our state of understanding, depend 
on a number of characteristic variables sufficiently small to make mathe- 
matical methods of analysis successful in predicting new results. The facts 
observed in nonexact sciences depend instead on such a complex inter- 
play of causes, that mathematics is often of limited help. 

Thus in this context, one can say that « exact » and « nonexact » are 
synonyms of « simple » and « complex », when these adjectives are used 
with the meaning specified above. 

In classifying some scientific activities as exact (or simple), we also yeld, 
often unconsciously, to a sort of hope. The hope that many, possibly 
all, natural phenomena will eventually be mastered using mathematical tools. 
This hope runs all along the history of science, from the Greek philosophers 
who believed that the Gods were thinking in terms of arithmetic and geometry, 
to Newton and Einstein who tried to unify physics, astronomy, and cosmology 
under common laws, and to present-day scientists who use quantum theories 
to interpret all physical and chemical phenomena that fall under our experience. 



82 G. Cocconi 

In physics, we still hope that relativistic quantum mechanics and electro- 
dynamics will be able to give a complete description of matter, from solid 
state to small and big molecules, from atoms to nuclei, from nucleons to 
subnuclear particles, i.e. hadrons, leptons, and their combinations made 
manifest by the study of the most energetic collisions produced either arti- 
ficially or by means of cosmic rays. We still hope that, once a certain degree 
of understanding is reached in these fields, also the problems one meets in 
other fields, even in the nonexact sciences, will slowly fall under the powerful 
methods of analysis developed for the more exact disciplines. It is easy to 
find in the history of science several good examples of this trend. 

However, there are also cases where disciplines believed to belong to the 
« simple » class were later found to be « complex ». For the ancients, the 
animal and the vegetable worlds on Earth appeared as the products of a single 
act of creation, while now we know how complex the path of evolution must 
have been. The situation is similar for geology and astronomy, even through 
many still hope that a simple « big bang » is at the origin of the whole so- 
called Universe. These are several branches of science whose first principles 
we now understand much less than our predecessors thought they did. 

The reason why I am talking about these matters here is that at the present 
moment several of the scientists involved in what is considered one of the 
most fundamental fields of research, the field of high-energy physics are begin- 
ning to ask themselves whether the complexities that become more and more 
manifest as the properties of matter are analyzed with higher spatial and 
temporal resolution could lead this branch of fundamental physics to a point 
where it becomes impossible to pretend that this discipline can be classified 
as exact. 

Before further elaborating this point, I will try to qualify the meaning of 
the term « nonexact » attributed to a branch of science. A concise definition 
is the following: Complex disciplines are those obeying laws that evolve 
historically. 

One must be aware of the fact that what makes our definition of complexity 
defy mathematical analysis is the word « historical ». This is because history 
contains the idea of the « arrow of time », a concept absent in all fundamental 
laws of physics that can be expressed mathematically. 

The classical example of historical evolution is that postulated by Darwin 
in order to understand the origin of life on Earth. We now know that all 
history of living organisms progressed via a string of processes, each a con- 
sequence, a chance consequence, of the previous one, and that a modifica- 
tion introduced in a step by a physical or chemical reaction had consequences 
that depended on the preceding history. In biological evolutions there is 
no means of going backwards to recreate the conditions that were present 



The role of complexity in nature 83 

at an earlier stage. This is the reason why the laws that govern life on Earth 
today are different from those that governed it yersterday and also from 
those that will govern it tomorrow. 

It is illuminating to realize that in order to give an « exact » reason for 
the concept of the unique direction of time it is necessary to bring into the 
picture an evolving system. In physics, there is no way of understanding the 
existence of the arrow of time if one remains within the framework of the 
fundamental laws of quantum mechanics, which are time symmetric. 

As pointed out by Gold, time asymmetry is created by the fact that the 
space around us acts, for the energy radiated into it, as a cold sink. Energy 
emanated from any reaction is lost into space, never to be returned. Time 
points in the direction corresponding to energy dump, a direction that is de- 
termined by the way in which the world around us evolved and is evolving. 

In our context, the most important peculiarity of evolution is that, in the 
complex situations where it works, it has the possibility of creating at each 
instant of its development the laws that govern it. 

For example, the way in which living matter organized itself in the early 
reducing atmosphere present on the Earth, was very different from the way 
in which it evolved later when the atmosphere became oxidizing because of 
the existence of life. 

It is this possibility of self-regeneration, coupled with the chance that 
enters in each step, that makes historical evolution so powerful in creating 
new possibilities, actually new laws. 

Now, the question is the following: can this kind of insight that we 
have gained from the nonexact disciplines be of help in developing further 
understanding of the « exact and simple » world of physics ? 

What is the consequence of evolution, of the arrow of time, in establishing 
the fundamental laws, the exact and simple laws of quantum physics ? Those 
who think that this way of arguing is sacrilegious may indeed be right; how- 
ever, they should be reminded that, when the arrow of time was not recognized 
to be operating in the biological sciences, the living world was considered 
static, dominated by fixed laws, apart from an act of creation that defied 
definition. And it made sense. 

In physics now we are in about the same situation; we have some fun- 
damental immutable laws to which all mechanical facts obey — conservation 
of energy, conservation of charge, constancy of velocity of light, etc., etc., 
and far back in time, ten billion years ago, some big event that put all this 
world into being. We accept the existence of an arrow of time, but we relate 
it to an inevitable emptiness of the space around us. Could it not be that 
this static conception is wrong, as was that of the biologists before Darwin ? 

In the following pages I will try to show that such a possibility exists, and 



84 G. Cocconi 

that complexity, far from bringing confusion, could in fact open vistas that 
our present culture has some tendency to dismiss. 

2. - Having stated our aims, let us look at the causes of complexity and 
try to establish a correlation between complexity and the property of the 
simplest phenomena of which complexity is built. It is found that outstanding 
correlation exists between complexity and the sign of the energy balance of 
the elementary reactions. Complexity invariably arises when the simplest 
reactions become endothermic, i.e. when energy has to be supplied to the 
evolving medium. The worlds of molecules, of atoms, of nuclei, as well as 
that of planets, stars and galaxies, are « simple » worlds only when exo- 
thermic reactions take place in them. 

Consider the case of nuclei. The building of heavier nuclei starting from 
lighter ones is exothermic only up to iron, then becomes endothermic. if the 
Coulomb repulsion is not circumvented. As a consequence, the formation 
of nuclei up to iron is easily achieved in the laboratory via collisions of two 
bodies and in nature through the slow burning of light nuclei in stars, as in 
the sun. However, the formation of heavier nuclei, like lead, uranium and 
beyond, asks for a lot of ingenuity on the part of nature. We now believe 
that it can be achieved through the very intense neutron bombardment 
(to overcome Coulomb repulsion) that can take place in rapidly evolving 
matter, but it is not yet clear whether these conditions can be found in a 
collapsing supernova or in the big bang (again!) that presumably took place 
ten billion years ago. In the laboratory, the heaviest nuclei have been ob- 
tained, so far, only via the bombardment of already existing heavy nuclei; 
never by bringing light nuclei together. 

Also the astronomical world is simple, beautifully simple, only when 
bodies at great distances from one another, placed there by a still unknown 
mechanism, slowly dissipate their gravitational energy to form planetary 
systems, streams of stars rotating into a galaxy, or nebular matter on the 
way to condensing into proto stars. The endothermic process that brings 
all this into being is certainly not simple, be it the initial big bang or the con- 
tinuous creation of matter of the steady state. 

An even better r story is presented by molecules, whose behaviour we 
are able to follow without interruption, from the simplest exothermic two- 
body reactions of inorganic chemistry, to the great organic constructions 
kept alive by the continuous flux of energy that, on Earth, comes from 
the Sun. 

These examples, besides illustrating how complexity needs an energy 
source, also show that, for building elaborate structures, it is necessary 
to have a medium where many-body reactions can take place within times 



The role of complexity in nature 85 

short in comparison with the disintegration lifetimes. On Earth, the molecules 
had the opportunity of arranging themselves into the long chains that even- 
tually led to DNA, in the warm broth imagined by Oparin, where the products 
of the reactions ocurring in the atmosphere and on the surface could meet 
and react; a process that is going on even today, within the living organisms, 
and is made manifest by mutations. 

These considerations are of importance for high-energy physics because the 
most characteristic property of elementary particles is that they are all produced 
via endothermic reactions. In classical nuclear physics the energetic particles 
produced by accelerators were used either to make stable structures or to 
overcome Coulomb repulsive barriers in order to add some nucleons to already 
existing nuclei, all exothermic processes. Today in high-energy physics the 
accelerators are used in a different way, namely to supply to the known stable 
particles the energy necessary to build more complex, heavier states, which 
require energy to be created. The so-called elementary particles obtained in 
this way, mesons and baryons, are thus the product of endothermic reactions, 
are heavier than the sum of the masses of their constituents and, consequently, 
have in general very short disintegration lifetimes, down to 10~ 22 s. Though 
the number of these particles is already quite large— more than one hundred — 
and their masses are as high as about three proton masses, we do not have 
any indication that particles of much higher mass cannot exist, albeit for short 
times. The present limitation in the number of particles reflects only the 
present maximum energy of the accelerators. High-energy physicists are 
convinced that as soon as larger accelerators are built, particles of larger 
masses will be discovered, having disintegration lifetimes not shorter than 
those already known today. 

In these circumstances, it becomes legitimate to ask whether situations 
could occur capable of bringing to the most fundamental branch of physics 
developments resembling those so far observed only in nonexact sciences, 
the developments produced by complexity. 

Thus, there are similarities between the world of elementary particles 
and that of molecules; both depend on the occurrence of endothermic pro- 
cesses and for both there is the possibility of building complex structures 
using simpler building blocks. How far one can build with elementary par- 
ticles is not known, but so far no limits have been felt, either. Actually, 
as long as we create particles as we do now, one by one, by bringing 
together two simpler particles and hoping that by hitting each other at 
great speed they will somehow coalesce, there is a drastic limit to the com- 
plexities that can be built. In the case of molecules, the progress would have 
been very limited indeed if chemists could produce compounds only by 
shooting together a pair of atoms or simple molecules. The building of 



86 G. Cocconi 

more elaborate structures requires the myriad of collisions taking place 
in a gas, in a solution, and, very important, the presence of catalysts that 
can keep alive intermediate steps. One needed the primitive oceans and 
billions of years in order to have the chance of creating the molecules of life. 
Of course, one also needed the practically inexhaustible possibilities of the 
carbon bonds, but can we say now that equally far reaching possibilities do 
not exist in structures composed of elementary particles ? 

This doubt makes it legitimate to speculate in which circumstances the com- 
plexities of elementary particles, if virtually possible, could become manifest. 

In the world as we know it today, it appears that the spontaneous evolu- 
tion of matter we are talking about can be induced only by gravity. Gravity is 
the only known force that does not saturate, i.e. does not cease to operate 
when many field sources are brought together, contrary to what happens for 
electromagnetic forces, weak coupling forces and molecular forces. The pull 
of gravity steadily increases as more and more matter is brought together. 
Just as nuclear reactions start when enough matter is condensed in a star, it 
is conceivable that the exothermic reactions of hadronic and leptonic matter 
will fully develop their possibilities only when large enough masses will com- 
press matter to the point where the characteristic energy per particle is well 
above the giga electronvolt. In matter consisting of baryons compressed to de- 
generacy by gravity, the characteristic momentum per particle kpc ~ 10 3 - 8 @*eV, 
where q is the density in grams per cubic centimeter. Already at the density 
of neutron stars (g c± 10 15 gcm~ 3 ) momenta up to ~ 1 GeV are available; 
these are also the densities of matter in nuclei. The places where these densities 
can be exceeded are those where matter condenses even further: in the col- 
lapsing neutron stars and possibly in galactic centres. It is not surprising 
by chance that at present we have no way of telling what the behaviour of 
matter is in these bodies. 

In the laboratory, present knowledge about the interactions of nuclei with 
giga electronvolt particles derives exclusively from the study of two-body in- 
teractions and it can be easily predicted that much more complex situations, if 
they exist, will not be observed as long as we continue to use this simple channel. 

A practical way of partially circumventing this limitation, short of creating 
the concentration of energy possible for astronomical bodies, is perhaps to 
use the collisions between particles and nuclei, and between nuclei and nuclei 
at effective energies (i.e. centre-of-mass energies between composite structures) 
of several giga electronvolts. Until now, particle-nuclei interactions have 
been studied at high energy only to gather information about the so-called 
coherent or diffraction dissociation reactions, where the nucleus acts as a single 
particle. The complex reactions we are looking for should be observed in the 
incoherent background, where the complex structures built by the collision with 



The role of complexity in nature 87 

the first nucleon have the chance of interacting further before leaving the nucleus. 

With the present accelerators, the energies available are not yet suffi- 
ciently high. Only with the new generation of 100 to 500 GeV accelerators 
will one obtain second-order collisions within the same nucleus with total 
cm. energies of many giga electronvolts. The best conditions in this respect 
will undoubtedly be obtained with colliding beams of nuclei. 

Large Lorentz factors for the complex structures built in these collisions 
are necessary not only to make available the energy needed for carrying out 
the endothermic reactions, but also for bringing together these unstable 
structures before they decay. Even if widths of the order of a few 100 MeV 
continue to be characteristic, mean decay paths equal to nuclear dimensions 
demand Lorentz factors larger than ten. 

3. - It is appealing to think that in the realm of high energies, situations 
could develop similar to those possible for molecules, and that subtle and 
apparently insignificant details of some interactions could have unimaginable 
and radical consequences in the historical evolution of matter. 

Clearly, the only justification today for this kind of science fiction is the 
observation of a great variety of endothermal high-energy reactions, a phe- 
nomenon that has some parallel with the molecular case. 

It should be realized that if such a possibility really exists, our concep- 
tions of the physical world would be greatly affected. 

The immutable laws of physics could become as « ephemeral » as those 
of organic life, immutable only for observations limited in space and in time, 
and even more exotic, the evolution of these laws would depend on history, 
a history that has followed a path that, to a great extent, must have been 
determined by chance. Seen from this point of view, even the Heisenberg 
uncertainty principle could be considered a temporary consequence of laws 
establishing themselves in an undeterminable manner. 

Another kind of life, the life of the physical world, would then be developing 
around us, in parallel with that we are accustomed to call the real life, that 
on Earth, of the organic world. 

Is the history of organic life a subset of the more general history of the 
physical world ? Has the evolution of organic life the possibility of interfering 
with the evolution of the physical laws — and vice versa! 

I realize that I have carried the argument to its extreme consequences and 
beyond, into the metaphysical sphere. But to me, it seemed useful to emphasize 
how complexity can have peculiar ways of manifesting itself through the 
possibility of developing momentous consequences from details apparently 
of no importance. 

The high-energy phenomena, as we know them now, seem to have prop- 
erties that could give rise to complexities of this kind. 



Channeling of Ultrarelativistic Charged Particles 
in Crystals. 



B. Ferretti 

CERN - Geneva 



When very high energy charged particles (usually electrons) or gamma 
rays pass through crystals, the coherent interaction with the lattice gives rise 
to several interesting phenomena, which have been studied both theoretically 
and experimentally in recent years [1]. 

There is an aspect of these phenomena, which until now has not been 
investigated experimentally, because it becomes important at energies higher 
than those actually at hand, and which, as far as I know, has not received 
either much theoretical attention, with the exception of a stimulating paper 
by Schiff [2]. We have in mind the fact that, in appropriate circumstances, 
the interaction between lattice and charged particles becomes so strong that 
the Born approximation, usually quite valid in the electrodynamic case at 
very high energies, has to break down completely [3]. This breakdown can be 
attributed to two circumstances: First, the fact that a description of the 
electrons wave functions by means of plane waves becomes very inadequate; 
and, second, the fact that the coherent emission of many quanta, or even a 
« coherent cascade » may become important. 

The resolution of the problem of finding a more adequate solution for the 
motion of very high energy electrons in crystals is preliminary dependent 
upon the investigation of the « coherent cascade ». This short note contains 
a few results obtained in trying to solve this preliminary problem. 

What we have in mind to discuss particularly is the possibility of « chan- 
neling » of very high energy electrons, or, better, positrons, around certain 
directions in a crystal. 

We shall consider, for simplicity, a cubic lattice, having the principal 
crystallographic axes along x, y, z and a particle traveling around a direction 
making a very small angle with the x axis. More precisely, if p x , Py, Pz are 
the component of the momentum of the incoming (free) particle, we consider 



Channeling of ultrarelativistic charged particles in crystals 89 

the case in which 

(1) #z>/V>/>z — °- 

Then, it might be shown [4] that the wave function of the electron inside 
the crystal can be suitably approximated by: 



(2) 
where 



4 v 

= A exp [ipx] y E ,* E n z) exp i V5 JVE'-V(yz') dz' J Xa (?) , 



2 

E' = E-p-^- c=l;K=l, 
2p 



E is the energy of the particle, p is the component of the Bloch momentum 
along x (p c^px), m is the mass of the particle, V(y, z) is the average of the 
potential of the crystal, taken along the classical path in a cell, and the func- 
tion Xa( z ) satisfies the equation 

(3) xte) + 2p(E a -VWi)Xa = 0> 

where V(z) is the average of V(x, y, z) in the xy plane. Inside the crystal, 
in place of eq. (1) we shall have 



(1') p^V2pE'^>V2pE a . 

If a is the pitch of the lattice, V(z) might be quite well represented by 

(4) V(z) = F (exp [- zfs ] + exp [- (a - z)/s ]) 0<z<a 

where 

s = aJZi, 

a is the atomic Bohr radius and Z is the atomic number of the element which 
is supposed to form the crystal. 

In the case of the positron (which is the most interesting for the reason 
that we shall see) the channeling is important when 

(5) Ea « V . 

In this case we see immediately, considering eqs. (2) and (3), that the 
positron can be trapped between two nets of equilibrium positions of the 
nuclei of the crystal. The planes % and n 2 of these nets are obviously orthogonal 



90 B. Ferretti 

to z. In this case it is quite clear that the particle might be channeled in a 
small neighbourhood of directions belonging to these planes. This channeling 
however can be disturbed by several effects, as the interaction with the elec- 
trons of the crystal, the effect of the zero point, and thermal motion of the 
atom, and so on. 

If E a is not much less than V , the most disturbing effect will be, in general, 
the interaction with the zero point and thermal motion. It is this effect which 
we have particularly tried to estimate. 

To arrive at expression (2) for the wave function of the positron one has 
essentially to use the fact that the potential of the lattice is periodic, and that, 
for the particular solution which we are considering, the frequencies of the 
components of the forces along x and y are very different. In this way, for 
instance, the forces along y can be considered as an adiabatic perturbation 
for the motion along x. 

These circumstances allow us to separate approximately the variables 
in the solution of the equation of motion, and to arrive at expressions (2) 
and (3). 

If. however, one takes into account the zero point motion, the perturba- 
tion due to the displacement of the lattice atoms will not be almost periodic, 
and consequently the separation of the variables will be no longer pos- 
sible. 

A preliminary estimate, however, shows that if condition (5) is satisfied, 
the effect of the atom motion is very small, in such a way that it can be con- 
sidered as produced by a small perturbation AK This perturbation A V will 
be a functional of the operators u x (x n , y m ), u 2 {x n -,y m ) which describe the 
displacement of the two nets ti 1 , tc 2 . (x n and y m are the co-ordinates of the 
equilibrium positions of the atoms belonging to n^) (*) 



1 v ■ 

=^,— ? (exp [iqx]a qj + exp [— iqx]a qi ) , 



VlNMv <»q 

where M is the atom mass and TV is the number of atoms for unit voume. 
Only the atoms, however, which are nearest to the classical trajectory can 
contribute appreciably to AV. For a given classical trajectory and for any x n 
there will be in general one (or in exceptional cases, that we shall disregard, 
at most two) of such atoms belonging to ji x ; and similarly, for any x' n , one 
belonging to n 2 . The « effective » displacement operators u x and « 2 therefore, 
might be considered as functions of a parameter only. Furthermore, the 



(*) a , and a Qi in the « x expression are the usual annihilation and creation operators 



Channeling of ultrarelativistic charged particles in crystals 91 

correlations of the orthogonal component of the displacement will not be 
important for determining AV when eq. (5) is valid. 

For these reasons we might use as well a two-dimensional, instead of a 
three-dimensional lattice, for studying the effect of the random motion of 
the atoms on the channeling. The wave equation then will be 

(6) (~ + ^) V>(*. z) + [£ 2 - m 2 - 2E(f(z) + AK(*, z))] y(x, z) = 0, 

where 

,r,^ K^r *r I \~ z\u-Xx) f a — zl w 2 (x)\ 

AV is obtained by expanding the potential V in powers of the displacements 
% and w 2 . 

The solution will be written in the form 

(6") y> = if a + Ay , 

where 

exp [ip a x] 
Wa = — —=—% a {z), 

A W=2,c b (x)y b , 

E =p a + AE a =p b + AE b , 

L being a normalization length. 

We can then solve our problem in the Born approximation with respect 
to the perturbation AV, in order to find the probability of transition for 
unit path, from any state ip a to any state ip b . This probability of transition 
is given in the usual form 

(7) P a M, "2) = 2?r| (y a |A*> 6 )| 2 e , 

where q = L/2ti is the density per unit energy of the states exp [ipx]fVL. 
We are of course interested in the average value (with respect to the motion 
of the atom) of this probability transition 

0') P ab = <\(y a \AV Wb )\*yL. 

The evaluation of <|(^JAK^ & )| 2 > can immediately be performed, and one 



92 B. Ferretti 

obtains : 



<|WAF n )|«>=^ 



o i r . r z i j i 2 
rg!J^ exp [-d^ d 1 ■ 

o 
1 . y J_ || sin(|/(/7 & — /? g )+gliL) 



2NM 



i(Pb-Pa) + tf| 



1 



1 — exp [— coJkT] 



+ 



+ 



sm(\i(p b —p a ) — q\$L) 



\i(Pb—Pa)—q\ 



exp [— co q fkT] 
1 — exp [ — wJkT)\ ' 



where i is a unit vector parallel to x. 

Using the Debye approximation, in the case in which r=0 we obtain 



(8) 



P ab — 



71 

16 



^^-{M'fe) 



2 / t; \ 2 ^71d/u — |i? & — /? a | 



kTvM 

if \Pb—Pa\< — 



if 1^6— A»l> 



&r D 



(v is the sound velocity and T D the Debye's temperature). 

Before using the result (8) we have to discuss the consistency of this result 
and the conditions under which it is valid. The undisturbed particle classically 
travels between n x and ji 2 , oscillating along z. During a complete oscillation 

it travels along x a length x (x~p\dz/V2p(E a —y), where z is the clas- 

sical amplitude of oscillation I. 

For the consistency of our results it is necessary that 



(9) 



b 



The condition (9), together with the condition that the quantum numbers 
of % a and % b are not too small, is also sufficient. 

In fact, if we follow our particle for a path x > x, due to the random- 
ness of the perturbation V, the states will no longer be a coherent super- 
position as in (6"), but a statistical superposition of incoherent states. 

Let p b be the probability of any pure state tp b . Then, in place of the wave 
equation (6) we have to consider the diffusion equation 



(10) 



&Pa 

dx 



J,Pab(Pb- 
b 



■Pa)> 



Channeling of ultrarelativistic charged particles in crystals 93 

which may be considered valid with P ab = P ba given by eq. (8), until the 
condition (9) is respected for all the states for which p a is appreciably different 
from zero. 

We can give some order of magnitude: If 

p = 10 10 eV , 

V = 150eV, 

Fo/M = 0.6xlO- 9 , 

v =10" 5 , 

A:r D /^=3xl0 3 eV/c, 

a = 3 x 10~ 8 cm , 

2z 0a =0.6«, 

2z 0b = 0.75 a, 



one has: 



P aa x^lO~\ P ab x~lO-*, lP„ c x-10- 



Leaving the other data unchanged, and taking z 0a = 0.25«, the mean 
free path for the diffusion of a particle which is initially in the state a will 
be of the order of one centimeter or more. In these conditions the scattering 
against the electrons might become more effective in destroying the channeling 
than the zero point motion. Obviously, in the case of incoming electrons, 
on the contrary, the zero point motion will be always very effective in destroying 
the channeling, because they would be trapped around the equilibrium posi- 
tion net of the nuclei, and not between two nets of equilibrium positions as 
it happens for positrons. 

Returning to positrons, of a beam impinging on the crystal, one half will 
have a « collision parameter » such that z Qa < 0.25a. The condition on the 
incidence angle ■& with the xy plane is much more restrictive: for/? =10 GeV, 
# ~ 10- 5 rad. 

The resulting restriction in the phase space of the incoming particle is 
however largely compatible with the indetermination principle. 

One can now ask whether our results can be generalized to cases in which 
the condition (1') is not satisfied. One might investigate whether, for instance, 
the trapping can arise between parallel crystallographic planes different than 
those which we have previously considered. 

Now, an immediate generalization of the preceding analysis shows that 
a necessary condition for the smallness of the zero-point motion perturba- 
tion is that the distance between neighbouring planes which are supposed to 
trap the particle is great compared to VTJiSkTvM). This condition rules 



94 B. Ferretti 

out almost all the possibilities, with the exception of the planes for which the 
Mill indices can be very small numbers. 

When the very severe restriction for trapping are not satisfied, the scat- 
tering will not be very different in a crystal or in a corresponding amorphous 
material. 

Concluding, we remark that it is perhaps possible either to use the chan- 
neling phenomenon which we have discussed for collimating very high energy 
positive particles, if one can obtain suitable crystals, or to use the channeling 
for studying very rare imperfections in almost perfect crystals by means of 
very high energy positive particles. 



REFERENCES 



[1] A rather complete list of references about this topic can be found in the report by 
U. Timm: Coherent Bremsstrahlung of Electrons in Crystals, DESY, 69/14 (March 1969). 

[2] L. I. Schiff: Phys. Rev., 117, 1394 (1960). 

[3] B. Ferretti: Nuovo Cimento, 7, 118 (1950). 

[4] See, for instance, the notes of the lectures given in 1969 at CERN, about The Brems- 
strahlung in Crystals. 



Experimental Work on Coherent Scattering 
of High-Energy Hadrons by Light Nuclei. 

G. Fidecaro and M. Fidecaro 
CERN-Trieste High-Energy Group - Geneva and Trieste 



1. - Introduction. 

In the last few years, several measurements [1-19] have been performed 
in order to study the coherent scattering of high-energy hadrons from light 
nuclei. We intend to present here the results so far obtained, including those 
contributed, in the case of the deuterons, by the CERN-Trieste High-Energy 
Group. A summary is given in Tables I-IV. 

Various techniques (emulsions, bubble chambers, counters, and spark 
chambers) have been used, as expected from the fact that the cross-sections 
involved vary by some orders of magnitude when the momentum transfer 
to the target particle is increased. Typically, in the pd case, at 13 GeV/c, 
the cross-section decreases from 0.3 b/(GeV) 2 in the forward direction, to 
0.1 [xb/(GeV) 2 for \t\ = 1.8 (GeV) 2 . Moreover, a good precision in the deter- 
mination of the kinematics of the events is of great importance for this type 
of experiments: an accurate reconstruction of the events in fact, on one 
hand helps to select the collisions which leave the target nucleus in the fun- 
damental state, on the other hand it provides the /-resolving power required 
to search for a possible structure in the differential cross-section. 

All the experiments but one [3] cover a limited angular region in the 
forward or backward direction and, correspondingly, two different kinds of 
phenomena are studied. 

Concerning the forward scattering region, in the first place these experi- 
ments give information on the structure of light nuclei. The use of strong 
interacting particles which have a non negligible probability of colliding at 
least twice with nucleons when traversing a nucleus, allows one to obtain 
results complementary to those obtained in the case of electron scattering, 
where the scattering amplitude depends predominantly on the single-particle 



96 G. Fidecaro and M. Fidecaro 

Table I. - Coherent (elastic) scattering of high-energy hadrons by light nuclei: a) protons. 



Momen- Momentum 
turn transfer 

(GeV/c) (GeV/c) 2 



Technique used 



Authors 



cm. 

(degree) 



pd 



p 12 C 
p 16 Q 



1.70 
2.78 
4.85 
6.87 
8.89 
10.90 
2.78 



1.70 



12.8 



p 4 He 1.2 



1.7 



1.7 
1.7 



0.003 < — /< 0.2 



0.44 <-/<1.54 



0.026<-f<3.44 



0.2 < — /<!. 



0.007<-f<0.7 



Emulsion to record the slow 
recoil deuterons from a tar- 
get of deuterated polyethy- 
lene - AE/E~2.5^5% 



Scintillator telescopes for both 
the proton and the deuteron; 
magnetic analysis and TOF 
measurement in the d-branch 

a) p-branch: magnetic spec- 
trometer (wire spark cham- 
bers), AE = 3 MeV, and 
TOF measurement 

b) d-branch: range spectro- 
meter 

Scintillation counters and wire 
spark chambers, At It — 2% 

p-branch: scintillator range te- 
lescope, A£'<20MeV 



N. Dalkhazhav et 4.1-^ 15.6 



0.007< — f < 0.47 | p-branch : magnetic spectrome- 
ter (wire spark chambers), 
I AE = 3 MeV, and TOF 
measurement 
0.023 <-?< 0.208 
0.009<-*< 0.509 



al. [1], JINR 
Proton Syn- 
chrotron 



E. Coleman et 
al. [2], Cos- 
motron 

G. W. Bennett 
et al. [3], 
Cosmotron 



3.1^- 15.6 



2.4 
1.4 
1.4 
1.2 
29.0 



9.3 

12.6 

8.9 

8.2 

55.0 



10.0^170.0 



CERN-Trieste [4] 7.8^- 23.1 
5.4^ 58.4 



E. T. Boschitz et 
al. [5], Virgi- 
nia University 

H. Palevsky et 
al. [6], Cos- 
motron 



6.9^- 58.1 



6.0^ 18.0 

3.5^- 27.3 



density. This is equivalent to saying that by using a hadron probe it is pos- 
sible to study nucleon-nucleon correlations inside a nucleus [20]. 

There is also a specific interest which concerns the hadron-nucleon ele- 
mentary interaction. At high energy ( > 1 GeV/c) the probability of having 
a single hadron-nucleon elastic collision decreases strongly when the mo- 
mentum transfer is increased, while, as it will be seen later, the probability 
for a double collision decreases less fast; that is, it is easier to obtain a large 
transfer of momentum by means of two subsequent collisions than by means 
of only one ; and there exists a ^-interval for which the two probabilities are 
nearly equal. This region, in which the two scattering amplitudes interfere, 
is sensitive to the phase difference between the single and the double scat- 



Coherent scattering of high-energy hadrons 97 

Table II. - Coherent {elastic) scattering of high-energy hadrons by light nuclei: b) pions, kaons, deuterons. 





Momen- 


Momentum 












tum 


transfer 








c.m. 






(GeV/c) 


(GeV/c) 2 




Technique used 


Authors 


(degree) 


K+d 


3.65 


0.05 <- 


-?<0.90 


20 in. deuterium bubble cham- 
ber 


H. C. Hsiung et 
al. [7], BNL 


8.0- 


-33.5 




6.0 


0.03 <- 


-?<0.21 


81 cm deuterium bubble cham- 
ber 
Scintillator hodoscopes for n-d 


CERN-Saclay [8] 


4.5- 


-12.0 


7r-d 


2.01 


0.262<- 


-/< 0.878 


R. C. Chase et 


26.1- 


-48.9 




3.77 


0.282<- 


-f< 0.898 


angular correlation; TOF 


al. [9], Argonne 


18.1- 


-32.7 




5.53 


0.291 <- 


-/< 1.232 


window in the d-branch, 
sweeping magnets in 7t and 
d branches to decrease the 
background 




14.7- 


-32.6 




0.895 


0.165<- 


-/< 0.940 


Wire spark chambers for Tr-d 


CERN-Trieste [10] 


37.2- 


-99.0 




0.994 


0.17 <- 


-/<0.46 


angular correlation, At/t~ 


CERN-Trieste [11] 


34.8- 


-70.5 




9.13 


0.20 <- 


-?<2.3 


~ 2%; d-TOF correlated (or 


CERN-Trieste [12] 


9.3- 


-21.0 




13.0 


0.20 <- 


-f<0.57 


not) with pion angle keeps 




7.5- 


-12.5 




15.2 


0.20 <- 


-f<1.02 


down the background 




7.0- 


-15.8 


K~d 


3.0 


0.27 <- 


-/<0.175 


81 cm deuterium bubble cham- 
ber 


W. Hoogland et 
al. [13], CERN 


6.5- 


-16.5 


dd 


2.2 


0.05 <- 


-/<1.9 


Deuterium bubble chamber 


M. Bazin et al. (pre- 








7.9 


0.05 <- 


-/<0.7 




liminary results - 
private commu- 
nication) 







Table III. - Coherent {elastic and quasi elastic) scattering of high-energy hadrons by light nuclei. 





Momentum 


Momentum transfer 








(GeV/c) 


(GeV/c) 2 


Technique used 


Authors 


Pd 


1.29 


7.0xl0" 4 < — /< 8.0 xl0~ 3 


Magnetic spectrometer, with 


L. M. C. Dutton and 




1.39 


8.0xl0~ 4 <-f< 9.5 xlO- 3 


sonic spark chambers, 


H. Buan van der 




1.54 


9.5 x 10" 4 < - /•< 12.0 x 10- 3 


A^ = 0.5% 


Raay [14] 




1.69 


1 1.0 xl0- 4 <-*< 14.0 Xl0~ 3 








19.3 


1.6xl0- 3 <-r<0.1 


Magnetic spectrometer, with 
sonic spark chambers, 


G. Bellettini et 
al. [15a] 


p 6 Li 


19.3 


-/<0.1 


Aplp = 0.5% 


[15a] 


7 Li 


19.3 


-f<0.1 




[15a] 


Be 


19.3 


-?<0.1 




[15a] 


C 


21.5 


-/<0.12 




G. Bellettini et 
al. [15b] 



98 G. Fidecaro and M. Fidecaro 

Table IV. - Coherent (elastic) scattering of high-energy hadrons by light nuclei (backward). 



Momentum 

(GeV/cm) Technique used 



Authors 



(degree) 



Pd 



1.70 
2.03 
2.25 
1.41 
1.70 
2.40 

2.65 
1.37 
1.70 
2.78 
4.50 
4.25 



1.9 



1.7 



Scintillator telescopes for both the proton 
and the deuteron; magnetic analysis 
and TOF measurement in the d-branch 

Scintillator telescopes for both the proton 
and the deuteron; range analysis (op- 
tical spark chambers) in the p-branch; 
magnetic analysis in the d-branch 



Scintillator telescopes for both the proton 
and the deuteron; magnetic analysis 
(kp/p = ± 2%) and TOF selection in 
the d-branch. (The reaction studied is 
dp -> dp) 



E. Coleman et al. [2] 


152.1^120.0 


BNL-Cosmotron 


153.5^117.3 




154.0-M20.0 


N. G. Birger et al. [16] 


152 




151 




149.4 




147.5 


Yu. D. Bajukov et al. [17] 


160 




159.5 




162 




165.5 


Yu. D. Bajukov et al. [18] 


145 


ITEF Proton Synchro- 




tron (internal beam, 




CD 2 target) 




J. Banaigs et al. [19], 


174.5^180 


Saturne 





G. W. Bennett et al. [3], < 170 

BNL-Cosmotron 



tering amplitudes; if, as it happens generally at high-energy, both the am- 
plitudes are purely imaginary (high absorptive process), the interference is 
destructive and the differential cross-section goes down to zero (*). 

If the ratio between the real and the imaginary part of the scattering 
amplitude is different from zero, the cross-section will show a more or less 
pronounced minimum; one could hope to evaluate from the depth of this 
minimum the dependence of the real part of the scattering amplitude on 
the momentum transfer, which at high energy cannot be obtained in any 
other direct way [21]. This is a case in which nuclear physics comes to help 
elementary particle physics. 

It is perhaps worth recalling also that attempts have been made to in- 
terpret the forward scattering of hadrons by nucleons in terms of a model 



(*) A similar situation arises when one compares the probability of a scattering of 
order n, with the probability of a scattering of order (w — 1). This leads to a typical dif- 
fraction pattern. 



Coherent scattering of high-energy hadrons 99 

which treates the target particle as a composite system. The nuclear cross- 
sections show indeed a t- (or «-) dependence very similar to the one observed 
when the scatterer is an elementary particle [22]. 

Concerning the backward scattering region, the present tendency is to 
try models in terms of Feynman diagrams with exchange of baryons. How- 
ever, the phenomenology is in a much more rudimentary stage and the ex- 
perimental work still very much incomplete. 

The progress made in the field of hadron scattering by light nuclei comes 
both from development of the Glauber model [23] of multiple scattering 
which tends to explain satisfactorily the forward scattering, and from the 
new technical developments which have allowed experimental physicists to 
perform measurements of very small cross-sections, thus making possible 
the continuous comparison and the improvement of the theory. The exist- 
ence of these new techniques, in particular, will certainly shed light and in- 
duce progress in the phenomenology of backward scattering. 



2. - The forward scattering region. 

The formulae given below have been derived [23, 24] in the limit of the 
high-energy approximation and of very small forward angles, i.e., one as- 
sumes that the wavelength of the incident particle is small in comparison 
to its range of interaction and that the angles considered correspond to the 
angular region near the forward diffraction peak. 

In this approximation the elastic scattering amplitude for the case of two 
colliding particles is given by: 

(1) f(k\k) = ^Jexp ft*-* 1 )- A][l-exp[i*(*)]]d*A, 

where k and k 1 are, respectively, the initial and the final momentum of the 
incoming particle in the laboratory system, and b = UJ + |) is the impact 
parameter. The term %{b) is a complex phase shift which in the case of 
spherically symmetric interaction is related to the better-known phase shifts 
of the partial wave analysis through the formula 



m = x{^Y L ) = u l 



Formula (1) is valid for a spin-independent interaction of an arbitrary 
shape. It has been obtained by using the approximation P z (cos0)-»-/ o (6-V— ?,) 
where — t is the four-momentum transfer, and by replacing the sum over / 
with an integral over b. 



100 G. Fidecaro and M. Fide car o 

The next step is to consider a system of A particles bound to form a 
nucleus. The approximation is here made that the single nucleons are frozen 
in their instantaneous position r x , ...,r A during the time that the incoming 
particle goes through the nucleus. The generalization of expression (1) is 

(2) F{q) = A Jexp fa.A]d*A<v>|l-exp [ X (b, r lt ..., rj]|y>> , 

where q = k— k 1 and \q\ 2 = — t. It is here that the most critical hypothesis 
of the model arises; that is, the phase factor x(b, r i, •■•,r A ) is assumed to 
be the sum of the individual phase factors 

A 

X(b, r lt ...,rj) = ^XiiP—Si) , 
i-i 

sj being the component of r t along the incident beam. As a result, 

(3) F(q) = ~ Jexp [iq-b]d*bjd\ ... d 3 ^^*(r l5 ..., O^QS^)' 

{l-n^-^Jexpt-^A-j^/^Odv)]^!,....^). 

A 

If one expands JJ, the scattering amplitude F can be represented as a 

i-i 
polynomial in the hadron-nucleon scattering amplitude fj(q). This poly- 
nomial is interpreted as a sum of terms originated by multiple scatterings: 
first, second, ... order term corresponds to a single, double, ... scattering; 
the highest term is of order A, which is a consequence of the fact that the 
Glauber model takes into account the multiple collisions just by adding the 
phase shifts. 

Various attempts [25-28] have been made to improve the Glauber model 
by dropping some of the approximations. However, it appears to be a 
delicate matter to introduce new corrections; for instance, in the case of 
the deuteron, Harrington [29] has shown that in some cases the off-shell 
contributions cancel the sum of all the higher order terms. On the other 
hand it is a fact that the experimental values for the cross-sections obtained 
until now, with the exception of the ones in a very backward direction, are 
fitted in a reasonable way if the scattering amplitude is given by eq. (3), 
with (d<r/d£),.,.= |Fte)| 2 . 

In most of these fits [30] the hadron-nucleon scattering amplitude which 



Coherent scattering of high-energy hadrons 101 



appears in formula (3) has been parametrized as 



0V 



f(q)= 4^0 — *"*)exp 



as suggested by the available experimental data, where a is the hadron- 
nucleon total cross-section, and a is the ratio between the real and the 
imaginary part of the scattering amplitude. In some cases more accurate 
amplitudes were used, when available from phase-shift analysis of experi- 
mental data [27] or from extrapolation to high energy via finite energy sum 
rules [27, 31]. 

Concerning the wave function ip(r x , ..., r A ), the experimental information 
is rather accurate in the cases where A < 3, at least for values of r not too 
small, while for A>4 particular models have to be taken. 

Czyz and Lesniak [32] have computed formula (3) by using an inde- 
pendent particle model and a Gaussian dependence on r for the single par- 
ticle density. The nuclear scattering amplitude is well approximated by 



F(q) 



ik_ 
2ti 



(R 2 + 2£ 2 ) exp 



mo 



l) m -.(l 



/ay- 



\27z(RZ + 2pz)) eXP 



(R 2 + 2fi 2 )q 2 



4/ 



R being the width of the single-particle density distribution. This fomula 
shows that if the hadron-nucleon amplitude is purely imaginary (a = 0), 
the nuclear amplitude F{q) is also purely imaginary, and the double scat- 
tering term (j = 2) has opposite sign to the single scattering term (j =1), 
and half its slope ; for the value of the momentum transfer at which the two 
terms become equal (in absolute value) the cross-section goes to zero, as 
mentioned in Sect. 1. This happens, when 



t n = 



(R 2 /2 + £ 2 ) 



In 



£(*+'): 



In a similar way other (A — 2) minima would arise. If |a| ^ the dip will 
be filled up fast when |a| is increased, while the other parts of the curve are 
rather insensitive to this parameter. 



3. - The hadron-deuteron scattering. 

Among the experiments already mentioned, the hadron-deuteron scat- 
tering should be the simplest one from the point of view of the analysis, 



102 G. Fidecaro and M. Fidecaro 

because only two nucleons can be involved as scatterers. This advantage 
is partly counterbalanced by the fact that the deuteron is a spin-one par- 
ticle. For unpolarized deuteron targets, the observed cross-section is the 
average of that found for the three states of polarization [33]: 



(SKi. |<mW)|m>|i 



The operator F(q, S) is a linear functional F(q, S) = F x {q, S) + F 2 (q, S) of 
the form factor operator S, which is a linear combination of the scalar form 
factor S and of the quadrupole form factor S 2 . Operators F x and F 2 rep- 
resent the single and double scattering, respectively, 

*i(a, S) =A(q)S(q[2) + Mq)S(-q/2) , 
and 

^^=44HWf+^(f-*')+^(f-^(f-*')- 

-c I [/ 1 (f + ,')-/ 2 (| + <)][/ 1 (|-.)-/ 2 (|-^)]) d v, 

where f x and f 2 are the elastic scattering amplitudes for collisions between 
the hadron and particles 1 and 2 of the deuteron. The coefficient C\ is 1 
if the incident hadron has isospin \, and is \ for an incident hadron of unit 
isospin. 

By choosing, for instance, a polarization axis along q, one observes by 
developing the above formula that only the double scattering can contribute 
to the spin-flip transition Am = ± 2 term, so that the dip in the differential 
cross-section, mentioned at the end of the previous section, is missing; as 
regards to the nonspin-flip transitions, both the cross-sections that corre- 
spond to an initial state m = and m = ± 1 show pronounced minima 
brought about by the destructive interference of the single and double scat- 
tering amplitude. The position of these two minima occur at rather dif- 
ferent values of q since in that range of momentum transfer S 2 (q[2) is nearly 
equal to S Q (qf2); thus the single scattering term, in the m = 1 state, which 
is weighted by the factor (S — »S 2 ), decreases faster than in the m = state, 
where the weight factor is (S -\- %S 2 ). As a result in the case of unpo- 
larized deuterons, the differential cross-section in the interference region is 
not very sensitive to the ratio between the real and the imaginary part of 
the scattering amplitude; the differential cross-section is instead very sen- 
sitive to the quadrupole form factor and therefore to the cf-wave percentage 
included in the wave function of the deuteron. 



Coherent scattering of high-energy hadrons 



103 



10 p 



10 3 



■4? KV 



• 0.895 GeVfc 
A 9.0 GeV/t 
o 15.2GeV/c 

CERN -TRIESTE 




-t (GeV/c) 2 



0.5 



IjO 



1.5 



20 



2.5 



Fig. la. - Differential cross-section for rc-d elastic scattering at 0.895, 9.0, and 15.2GeV/c. 
The experimental data are from refs. [10] and [12]. The 0.895 curve (•) (ref. [27]) was 
computed by using the pion-nucleon amplitudes derived from the CERN phase-shift 
analysis. For the 15.2 GeV/c curve (o) (ref. [27]) the amplitudes were those of Barger 
and Phillips (ref. [21]). In both cases the Gartenhaus-Moravcsik deuteron wave function 
was used. For the 9.0 GeV/c curve (a), the scattering amplitude was parametrized as 
/fo) = (ifcff/4OT)(l — ia)exp[— 0V/2], with a p = 26.9 mb, a n = 25.3 mb, a p =-0.13, 
a n = — 0.23, /?* = fil= 8.5 (GeV) -2 ; the Humberston wave function was used for the deuteron. 



104 



G. Fidecaro and M. Fidecaro 



• pd-» pd 
12.8 GeV/c 
preliminary 
CERN-TRIESTE 

a pd-»-pd 
1Q9 GeV/c 
Kir ill ova et al 




1.0 

-t(GeV 2 ) 



Fig. lb. - Differential cross-section for p-d elastic scattering at 10.9 GeV/c (a) and 
12.8 GeV/c (•). The experimental data are from ref. [1] and [4]. For the nucleon- 
nucleon scattering amplitude it was assumed that a p = 39.2 mb, a a = 40.2 mb, 
a p = « n = -0.33, Pl = fil = 10 (GeV)- 2 ; for the deuteron the Humberston 
wave function was used. 



Coherent scattering of high-energy hadrons 105 

In the case of polarized deuterons, if q& is the position of the minimum, 
the cross-section at q^ is strongly sensitive to the real part of the hadron- 
nucleon scattering amplitude at the value q = q&jl and q = q&. 

In Fig. 1 we show a sample of data for pions and protons. There is no 
dip and the agreement with the theoretical curves is rather good. 

The coherent quasi-elastic scattering can be described in a similar way, 
starting from formula (3). The agreement with the experimental data (see 
Table IV) is again rather good. 



4. - The A>2 case. 

A similar analysis has been carried out for He and other light nuclei. 
Here the information exists only for protons and is rather limited. The 
experimental data for 4 He are compared in Fig. 2 with the curves foreseen 
by the model. At low momentum transfer the agreement is good for what 
concerns the size and slope of the diffraction peak as well as the size and 
position of the first minimum. 

The situation changes at larger angles. It seems that a better represen- 
tation of the data is obtained by decreasing the scattering amplitude with 
a change of the nuclear density, in such a way that the term of order n is 
decreased less than the term of order (n — 1). 

Because, roughly speaking, the multiple scattering amplitude is an ex- 
pansion in cr-r^f, where r tj is the distance between two nucleons in the nu- 
cleus, one can obtain the above effect by increasing the average separation 
of the nucleons themselves. This is the case when there are nucleon con- 
figurations which are preferred or if there is a correlation such that the two 
nucleons cannot approach each other beyond a certain limit. In this scheme 
Czyz and Lesniak [32] and Bassel and Wilkin [30] have modified the nuclear 
density function, given by the independent particle model. 

In Fig. 2 the results of Bassel and Wilkin are reported. They tried to 
fit the experimental data of ref. [6] either by using a double Gaussian as 
density function for the single particle, or by introducing a correlation 
function. The fit of these data (as well as of the data for e- 4 He elastic scat- 
tering) is good in the first case (continuous curve of Fig. 2), but not in the 
second one (*). 



(*) A critical analysis of the kind of interpretation has been made recently by Cro- 
mer [34]. 



106 



G. Fidecaro and M. Fidecaro 



10 



10^ 



> 
E10° 



D 



10"'- 



10" 2 =- 



10" 



n. ' 


i i i i i i i i i i i i 


1 : 




p- 4 He 


] 


"\\ 


1.7 GeV/c 


- 




Palevsky et al. 


~ 


= i\ 




'- 


■ v> 




- 




\ 
\ 


-3 


: \\\ 


\ 


Z 


\i 


\ 
\ 


- 


V 




- 


Z 




~ 




\W // \ •*>* ^i. N 






*T/. \ ^ s TL x 




— 


« \ V\\ 

\ N \ \ s 





I 


\ \V \ 


- 


- 


\ VN \ l\ 
\ w \ i«[ x 


- 


~ 


\ w^h* 


- 


E 


\ ^ r ^ • 


ft 


_( 3) 1 




\ N * N 


■LI 




- 


(D\ 


v 


■M : 










i 


1 1 1 \ 1 1 1 1 1 1 


\l 


1 -• 



0.5 



1.0 



1.5 



-t (GeV/c) 



Fig- 2. - Differential cross-section for p- 4 He elastic scattering at 1.7 GeV/c. 
The experimental data are from ref. [6]. The curves are from ref. [30], and 
were computed by assuming for the single-particle density a Gaussian distri- 
bution of width R 2 = 1.87 fm 2 ( ), or a double Gaussian distribution ( ); 

for the nucleon-nucleon scattering amplitude it was assumed that a = 44 mb, 
a = - 0.3 and ^ = 5.4 (GeV/c) 2 (- - -), or = 40.4 mb, a = - 0.5 and 
fP = 54 (GeV)~ 2 ( ). Curve (1) represents the contribution of single scat- 
tering (impulse approximation), curve (2) single plus double, etc. 



The results of a similar analysis for the 4 He data at 1.2 GeV/c [5], and 
for 12 C and 16 at 1.7 GeV/c [6], are reported in Figs. 3, 4 and 5. 

For the heavier nuclei, A^>1, the scattering amplitude which results 
from the sum over the various terms tends to the same form that one would 
find by using an optical model of the nucleus; this, in fact, is already true 
for the 16 case [35]. 



Coherent scattering of high-energy hadrons 



107 



5. - The backward region. 

The backward scattering has been studied only in the pd case, and in 
a rather limited momentum interval. The important experimental fact is 



10 



p- 4 He 



1.2 GeV/c 
Boschitz et al. 




J I I 



I I W I 



0.5 
t (GeV 2 ) 



1.0 



Fig. 3. - Differential cross-section for p- 4 He elastic scattering at 1.2 GeV/c. 
The experimental data are from ref. [5]. The curves are from ref. [3>5b] and 
were computed by assuming for the single-particle density a Gaussian distri- 
bution of width R 2 = 1.56 fm 2 ( — ) or the one obtained from the e- 4 He elastic 

scattering ( ); for the nucleon-nucleon scattering amplitude it was assumed 

that a = 39 mb, a = — 0.43 (- - -), and a = — 0.5 ( ), £ 2 = 4.3 (GeV/c)" 2 . 



108 G. Fidecaro and M. Fidecaro 




P 12 C 

1.7 GeV/c 
Palevsky et al 



0.2 
-t(GeV 2 ) 



0.3 



0.4 



Fig. 4. - Differential cross-section for p- 12 C elastic scattering at 1.7 GeV/c. The 
experimental data are from ref. [6]. The curve is from ref. [356] and was com- 
puted by assuming for the single-particle density a Gaussian distribution of 
width R 2 = 2.5 fm 2 , as obtained from electron scattering; for the nucleon- 
nucleon scattering amplitude it was assumed that a = 44 mb, a = — 0.28, 

/S 2 = 5.4 (GeV/c)- 2 . 



the strong enhancement of the cross-section toward the largest angles, as 
it is shown in Fig. 6. 

In order to extend to the large angles the mechanism that de- 
scribes forward scattering, let us refer to the pictorial representation of the 



Coherent scattering of high-energy hadrons 109 



p l6 
1.7 GeV/c 
Palevsky et al. 




n 



0.1 



Q2 
- 1 (GeV/c) : 



0.3 



0.4 



Fig. 5. - Differential cross-section for p- le O elastic scattering at 1.7GeV/c. 
The experimental data are from ref. [6]. The curve is from ref. [356] and was 
computed by assuming for the single-particle density a Gaussian distribution 
of width R 2 = 2.92fm 2 ; for the nucleon-nucleon scattering amplitude it was 
assumed that <r p = 47.5 mb, cr n = 40.0 mb, a = — 0.4, /S 2 = 4.7 (GeV/c)- 2 . 



Glauber model that is indicated in Fig. 7. If the graphs are interpreted as 
Feynman diagrams and certain approximations are made, one obtains the 
same values for the cross-section as those obtained from the multiple scat- 
tering model. 



110 G. Fidecaro and M. Fidecaro 
10 2 r ■ 1 



Pd 

Coleman et al. 

• 1.7 GeV/c 
o 2.0GeV/c 
A 2.25GeV/c 



1 1.7GeV/c 




cms 



Fig. 6. - Differential cross-section for p-d elastic scattering in the backward 
direction at 1.7 (•), 2.0 (o), and 2.25 (a) GeV/c. The experimental data are 
from ref. [2]. The curves are from ref. [38] and were computed from the triangle 

diagram of Fig. le. 



There have been attempts [36, 37] to use this technique for the backward 
scattering, but the computed cross-section is an order of magnitude lower 
than the experimental data. 

An attempt [18] to explain the data on the basis of the exchange diagram 
of Fig. le has also been unsuccessful, because the cross-section does not 
decrease fast enough when the momentum of the incoming proton is increased. 



Coherent scattering of high-energy hadrons 



111 



a) impulse approximation c) double charge exchange 

scattering 
P r~\ P P r \ n r\ P 





b) elastic double scattering d) inelastic double scattering 





e) one-nucleon exchange 



-o 



f) triangle graph 
Pi ^ d 2 



O 




^ k(pion) 
n D„ 



Fig. 7. - Graphs describing the pd elastic scattering in the forward (a), b), c), d)) 
and in the backward (e), /)) direction. 



Very recently, Craigie and Wilkin [38] related the proton-deuteron back- 
ward scattering to the case of a pp collision in which a fast deuteron is 
produced together with a slow pion; the pion (dashed line in Fig. If) is 
afterwards absorbed by the spectator nucleon. The predicted cross-sections 
are too low, but the angular dependence is in good agreement with the ex- 
perimental data. 



6. - Conclusions. 

The experimental results reported in this paper tend to indicate a fairly 
good agreement of the experiments with the Glauber model of multiple 
scattering. As mentioned in the introduction, this progress has been made 
possible by new technical developments. These can still be exploited for 
more refined measurements in the forward scattering region and for new 



112 G. Fidecaro and M. Fidecaro 

measurements at larger angles, including the backward scattering region. 
In particular, it seems to be of great interest to perform experiments of this 
kind on polarized deuterons, as one could check the model in detail through 
simplified experimental conditions. Unfortunately the technique of polarized 
targets has not yet reached such a point of refinement. 

The success so far obtained does not imply, however, that the theory is 
already perfect: much more systematic work of comparison with experi- 
ments is needed in order to clarify the exact meaning of the various assump- 
tions and approximations, and to find out the limit of validity of the pre- 
sent ideas. 



REFERENCES 



[1] N. Dalkhazhav et ah: Soviet J. Nuclear Phys., 8, 196 (1969). 

[2] E. Coleman, R. M. Heinz, O. E. Overseth and D. E. Pellet: Phys. Rev., 164, 

1655 (1967). 

[3] G. W. Bennet, J. L. Friedes, H. Palevsky, R. J. Sutter, G. J. Igo, W. D. Simpson, 

G. C. Phillips, R. L. Stearns and D. M. Corley: Phys. Rev. Lett., 19, 387 (1967). 

[4] F. Bradamante, G. Fidecaro, M. Fidecaro, M. Giorgi, P. Palazzi, A. Penzo, 

L. Piemontese, F. Sauli, P. Schiavon and A. Vascotto: presented at the Lund 

Intern. Conf. Elementary Particles, 25 June - 1 July 1969. 

[5] E. T. Boschitz, W. K. Roberts, J. S. Vincent, K. Gotow, P. C. Gugelot, C. F. 

Perdrissat and L. W. Svenson: Phys. Rev. Lett., 20, 1216 (1968). 
[6] H. Palevsky, J. L. Friedes, R. J. Sutter, G. W. Bennet, G. J. Igo, W. D. Simpson, 
G. C. Phillips, D. M. Corley, N. S. Wall, R. L. Stearns and B. Gottschalk: 
Phys. Rev. Lett., 18, 1200 (1967). 
[7] H. C. Hsiung, E. Coleman, B. Roe, D. Sinclair and J. Van der Velde: Phys. Rev. 

Lett., 21, 187 (1968). 
[8] P. Fleury: Thesis, January 1967 (unpublished). 

[9] R. C. Chase, E. Coleman, T. G. Rhoades, M. Fellinger, E. Gutman, R. C. Lamb, 
F. C. Peterson and L. S. Schroeder: Phys. Rev. Lett., 22, 1265 (1969). 
[10] F. Bradamante, S. Conetti, G. Fidecaro, M. Fidecaro, M. Giorgi, A. Penzo, 

L. Piemontese, F. Sauli and P. Schiavon: Phys. Lett., 28 B, 191 (1968). 
[11] F. Bradamante, S. Conetti, G. Fidecaro, M. Fidecaro, M. Giorgi, P. Palazzi, 
A. Penzo, L. Piemontese, F. Sauli, P. Schiavon and A. Vascotto: Lett. Nuovo 
Cimento, 1, 894 (1969). 
[12] F. Bradamante, G. Fidecaro, M. Fidecaro, M. Giorgi, P. Palazzi, A. Penzo, 
L. Piemontese, F. Sauli, P. Schiavon and A. Vascotto: submitted to the Lund 
Internat. Conf. Elementary Particles, 25 June - 1 July 1969. 
[13] W. Hoogland et al. (Sabre Collaboration): Nucl. Phys., B 11, 309 (1969). 
[14] L. M. C. Dutton and H. B. van der Raay: Phys. Rev. Lett., 21, 1416 (1968). 
[15a] G. Bellettini, G. Cocconi, A. N. Diddens, E. Lillethun, G. Matthiae, J. P. 

Scanlon and A. M. Wetherell: Phys. Lett., 19, 341 (1965). 
[156] G. Bellettini, G. Cocconi, A. N. Diddens, E. Lillethun, G. Matthiae, J. P. 
Scanlon and A. M. Wetherell: Nucl. Phys., 79, 609 (1966). 



Coherent scattering of high-energy hadrons 113 

[16] N. G. Birger et al: Soviet J. Nucl. Phys., 6, 250 (1968). 

[17] Yu. D. Bajukov et al: Soviet J. Nucl. Phys., 5, 236 (1967). 

[18] Yu. D. Bajukov et al: Soviet J. Nucl. Phys., 8, 199 (1969). 

[19] J. Banaigs et al: submitted to the Lund Intern. Conf. Elementary Particles (1969). 

[20] See also A. de Shalit: Comments on Nucl. Particle Phys., 3, 88 (1969). 

[21] See, for instance, L. Bertocchi: Comptes Rendus, Rencontre de Moriond sur les inter- 
actions electromagnetiques, 1967. However, in the last year a great effort has been 
made to get this information by fitting the available pion-nucleon data and through 
the sum rules technique, e.g- V. Barger and R. J. N. Phillips: Wisconsin preprint, 
April 1969. 

[22] See, for instance, T. T. Chou and C. N. Yang: Phys. Rev., 170, 1591 (1968). 

[23] R. Glauber: Lectures in Theoretical Physics (Wiley, Interscience, New York, 1959), 
Vol. 1, p. 315. 

[24] R. Glauber and V. Franco: Phys. Rev., 142, 1195 (1968). 

[25] L. F. Schiff: Phys. Rev., 176, 1390 (1968). 

[26] J. Pumplin: Phys. Rev., 173, 1651 (1968). 

[27] G. Alberi and L. Bertocchi: Nuovo Cimento, 63, 285 (1969). 

[28] G. Alberi and L. Bertocchi: Nuovo Cimento, 61 A, 203 (1969). 

[29] D. R. Harrington: preprint, Rutgers University (1968). 

[30] R. H. Bassel and C. Wilkin: Phys. Rev., 174, 1179 (1968). 

[31] C. Michael and C. Wilkin: Nucl. Phys., Bll, 99 (1969). 

[32] W. Czyz and L. Lesniak: Phys. Lett., 25 B, 319 (1967). 

[33] R. Glauber and V. Franco: Phys. Rev. Lett., 22, 370 (1969). 

[34] A. H. Cromer: Nucl. Phys., Bll, 152 (1969). 

[35a] R. J. Glauber: Proc. Int. Conf. High-Energy Physics and Nuclear Structure (North- 
Holland Publ., Amsterdam, 1967). 

[356] L. Lesniak and H. Wolek: Nucl. Phys., A 125, 665 (1969). 

[36] J. N. Chahoud and G. Russo: Nuovo Cimento, 49 A, 206 (1967). 

[37] L. Bertocchi and A. Capella: Nuovo Cimento, 51, 369 (1967). 

[38] N. S. Craigie and C. Wilkin: Elastic proton-deuteron scattering at large angles (to 
be published). 



The Determination of the Axial -Vector Coupling 
for Strangeness Nonchanging Currents. 

C. Franzinetti 

Istituto di Fisica - Universitd di Torino 

G. Stoppini 

Istituto di Fisica - Universitd di Pisa 



1. - Introduction. 

According to our present knowledge, weak interaction processes can be 
described using a phenomenological Lagrangian density of the type 

(1) & = -^J x j; + h.c., 

V2 

where 

(2a) J x =j$»+jf> + jp 

(26) JP = Wa(1 + 7 5 )Vve 

(2c) ffl = i^yxil + y 5 ) W 

are the local leptonic currents, associated with the electronic and muonic 
field, respectively. The hadronic current J^ has, in general, a more com- 
plicated structure: Following Cabibbo's theory, the strangeness conserving 
and strangeness nonconserving parts add up to give 

(3) /f)(+) = cos0 v Fj+> + cos0 A ^(/> + sin0 v F s ( +> + sin0 A ,4<+> , 

where V, A, are the vector and axial vector parts of the strangeness non- 
changing currents and V s ^ and A sX the corresponding parts of the strange- 
ness changing currents. The (+) suffix is a remainder of the fact that the cur- 
rent is a charge-raising operator (whereas its conjugate J* is a charge-lowering 



Axial-vector coupling 115 

operator). Cabibbo's angles VjA have the values [1, 2] 

6 y = 0.232 ±0.013 
d A = 0.250 ±0.01 8. 

G is the universal Fermi constant 

(4) G = (1.4350 ± 0.001 1) x 10" 49 erg cm 3 . 

The operation indicated by the asterisk is defined by the equation 

w) <> jc = Jjc » J 4: = J 4 > 

where + indicates Hermitian conjugation. The change of sign of the fourth 
component puts this component on the same position as the others with 
respect to transformation under charge- and G-conjugation operations. 

The strangeness-conserving vector current V x is supposed to behave like 
a « conserved current », i.e., to satisfy 

(6) hV* = 0, 

whereas the axial vector part is certainly not conserved. According to the 
«isotriplet hypothesis » V x , V*, and the isovector part of the electromagnetic 
current (l/e)/J' v form the three components of the same isotriplet. 

The most general matrix element of ./= V-\-A between two single nu- 
cleon states, satisfying the requirements of invariance of the Lagrangian den- 
sity under proper Lorentz transformation has 

(7) V x = iu p (Ayx ± faxr q^ ± / 3 qx) u n exp [— iqx] , 

(8) Ax = z'WpteiWs + gz<*3LrWs + gzqm)u n exp [— iqx] . 

The ft and g*'s are (in general complex) functions of the four-momentum 
transfer squared 

(9) q 2 = (p p -p Q ) 2 . 

If the conserved vector current hypothesis is valid, then 
(10a) f 3 (q*) = 0. 

Moreover, if the isotriplet hypothesis is accepted, then 
(106) f x (q*) = F*™iq*) 

(10c) f 2 (q*) = - ^^ F**iq*) , 



116 C. Franzinetti and G. Stoppini 

where F ± and F 2 are Hofstadter's nucleon form factors normalized to unity 
for # 2 ->0; ju p and fi n are the anomalous magnetic moments of the proton 
and neutron, respectively in units of Bohr magnetons (^ = 3.69); and M 
is the nucleon mass M a = M p = M. 

Invariance under time reversal operation imposes 

whereas charge symmetry requires 

gi = g*, g2 = —gt, g 3 = —g$- 

Thus g x is real ; g 2 = 0, and g 3 pure imaginary. At variance with the 
corresponding form factors of the vector part, the g's are not normalized to 
unity when q 2 -+0. Putting 

G v = Gcos6 w , G A = Gicosd A , G p = G°cos0 p , 
lim gl (q*) = GJG y , lim gjq*) = i(G p /G w ) , 

the entire matrix element for J^ reads 

(11) <p|/| h )|n> = i^m [Fiyx-^FiOM + 

«(n) exp [— iqx] , 



c c 

^F A y x y 5 + i-^ 



+ ^ F A.nYs + i^FpWs 



where 

*i(0) = F A (0) = 1 . 



2. - The determination of G A in experiments at low momentum transfer. 

In the limit of low q% the matrix elements (11) reduces to 

(12) <pMn> = -/j «p Ya(G y + G A y 5 )u n . 

Equation (12) can be used to predict the values of relevant parameters and 
distributions accessible to experimental determination. From the total decay 
rate of the neutron, one deduces 

(13) [i|G v | 2 + !|G A | 2 P = (1.63 ± 0.02) x 10- 4 » erg cm 3 , 



Axial-vector coupling 117 

G w can be deduced independently from a pure Fermi -> transition. 
On the basis of seven transitions [3] ( 14 0, 26 A1, M C1, 42 Sc, 46 V, 50 Mn and 54 Co), 
the value of G w = (1.4149 ± 0.0022) x 10" 49 erg cm 3 was obtained. Thus 

(14) GJG N = 1.18 ±0.02. 

This is the best determination of GJG N for strangeness nonchanging 
currents. All other experiments, though not as precise, have given results 
consistent with eq. (14) (*). 



3. - The physical interpretation of the experimental result GJG y = 1.18. 

The physical interpretation of this result received a first contribution when 
Goldberger and Treiman discovered that 



(15) 



G v ~ M ' 



where g n xx is the renormalized pion nucleon constant and f„ the pion decay 
constant. 

Their deduction was based on complex arguments of dispersion theory: 
however it was later rederived starting from different assumptions and using 
simpler methods [4]. 

Let the pion field be defined by the equation 

(16) d x A x = c n <p n , 

c r , being a constant (**). The matrix elements of (16) between two nucleon 
states is 

(17) <pMa|i» = *W[2MG A F A (<? 2 ) + q 2 G p F p (q*)]u p y 5 u n , 
N being a normalization coefficient. For q 2 ->■ this reduces to 

(18) <p|M» = iN2MG A u p y 5 u a . 

We must now calculate the matrix element of the pion field between the 



(*) For a complete discussion on this subject see ref. [3] and T. D. Lee and C. S. Wu: 
Ann. Rev. Nucl. Sci., 15, 381 (1965). 

(**) We omit here isospin indices which are not essential. 



118 C. Franzinetti and G. Stoppini 

same states. Using the field equation 

(19) (P2-»ft<P n (x) = J w (x) 
we obtain 

(20) <pk(0)|n> = - -j-L_ <p| jW(0)|n> = 

D(q 2 ) 

= ~ a 2 _L m 2 A/2 ftrj^P ^W p y 5 « n , 

where q^ is the pion momentum and D{q 2 ) is, by hypothesis, a slowly varying 
function of q 2 in the interval 0># 2 > — ra 2 and accounts both for the re- 
normalization of the free propagator and the free vertex. The \/2 factor 
arises from the isospin pion-nucleon coupling. Then D(q 2 ) is intended to be 
normalized to 1 for q 2 = — m 2 , i.e. 

Z)(— /w 2 ) = l. 

At # 2 =0 we hawe, using (16), 

2MG A = -V2^fD(0)c n . 
mi 

We can deduce c n from the measured 7>decay rate. This is in fact deter- 
mined by the matrix element 

<0|i4j7U> = I^^ a <0|^|7U> 

and one obtains 

c n = y/2f n m% , 

where f n is the (experimental) decay constant of the pion (y/2f n ^ 
~133MeVxG v [5]). 
Thus 

(21) G A = ^™^ G v D(0) . 

If D(q 2 ) does not change appreciably moving from q 2 = — m 2 to q 2 — 0, 
then (21) coincides with (15). Putting the measured values of the constants 
in (15), one obtains 

GJG W ~1A. 



Axial-vector coupling 119 

The difference between this value and the experimental one (eq. (14)) 
is ■ — -15%. The origin of such difference may well be due to the (q 2 = — m 2 ^- 
_> q* = 0) extrapolation of the form factor D(q 2 ) (see Gell-Mann and Levy [4]). 

The hypothesis formulated in eq. (16) is often referred to as Partially 
Conserved Axial-Vector Current hypothesis (PC AC). It may be expressed 
in a different way, namely by stating that the matrix elements of the diver- 
gence of the axial-vector current satisfy unsubtracted dispersion relations and 
that the immaginary part of this amplitude is essentially determined by the 
one-pion pole at q 2 = — m^.. Thus, although no extrapolation of the form 
factor is involved here the result is not exact because of the somewhat arbi- 
trary neglect of the background contribution beginning at q 2 =— m 2 . 

A major step toward our understanding of the origin of the renormaliza- 
tion of the axial- vector coupling was made independently by Adler [5] and 
by Weisberger [6]. 

Adler's deduction which is based on a method proposed by Fubini and 
Furlan [7], starts from the following assumptions: 

a) That the pion field is defined by the equations 

SttJVJV 1 °V 

« i » being an isotopic spin index and C% an isospin Clebsch-Gordan coefficient. 

b) That the fourth components of the axial-vector currents satisfy 
Gell-Mann's equal-time commutation relations 

(23) [A\{x\ Ai(y)] x ^ Jt = - d(x- y )e m V*(x) 

and obtains the following equation relating G V /G A to the off-mass-shell pion- 
nucleon cross-sections : 

/G v \ 2 2M 2 1 f dW 2 . . , , . , X1 

(24) ' " (si) = ^ n) m=m ['°v* + p)— »- ff (*-p)] . 

The numerical evaluation of (24) requires some care since the off-mass- 
shell cross-sections are not the experimental ones. He carries out the extra- 
polation assuming that the 3-3 /?-wave resonance predominates and obtains 

GJG y = 1.24. 

Weisberger, using a different formulation which does not imply off-mass-shell 
extrapolations (nor the Goldberger and Treiman relation) obtains 

G A /G V =1.15 



120 C. Franzinetti and G. Stoppini 

in excellent agreement with the experimental value (14). Other methods to 
obtain (24) have been indicated (Adler [5], Weisberger [6] Fubini and 
Furlan [7]) which the reader may find in the literature quoted here. 

It is interesting, in this connection to mention that Weinberg [8] has 
obtained a sum rule which differs from (24) only in terms of the order 0(m 2 fM^,) 
starting from the Goldberger-Myizawa-Oehme sum rule for the pion-nucleon 
scattering lengths a%— a%. Turning the other way around one deduces a 
relation between aj — a% and GJG y . 



4. - The # 2 -dependence of the axial-vector coupling. 

4*1. Neutrino experiments. - The most direct way of investigating the 
dependence of the axial-vector coupling on q 2 is the study of elastic reactions 
produced by energetic neutrinos, i.e., processes of the type 

(25) v t +n->r+p 

(26) v t +p^t + +n, 

where t is a lepton and vt the neutrino associated with the corresponding 
lepton field. 

The theoretical distribution da/dq 2 =f(E Vi q 2 ), ZT V being the primary neu- 
trino energy, can be deduced from eq. (11). Neglecting terms proportional 
to the lepton mass, one obtains 

(27) (S); = S S* k [A(q2) ± xB ^ + x * c « ] ' 

the ± sign before B(q 2 ) referring to v and v reactions, respectively. Moreover, 
putting A/x = ^ p — /^ = 3.7 B.N.M. 

A = 4MV (^J (F 2 - F 2 ) + q 2 [fj + 4£^F x F t + F 2 ty + (^V^l 

(28) iB^F^ + ApF^q 2 

Experiments to measure the functions A(q 2 ), B(q 2 ), C(q 2 ), and hence F A (q 2 ) 
have been performed at CERN and Argonne National Laboratory, using 



Axial-vector coupling 121 

spark chambers and heavy liquid bubble chambers. The analysis was 
carried out assuming F 1 and F 2 were identical with the electromagnetic form 
factors; and F A (q 2 ) was represented by the parametric form 



(29) F A (q 2 ) 



(•+& 



with M A to be determined from fitting the experimental data. The results are: 

a) at CERN: 

Spark chamber exp. [19] M A = 0.65±£ff GeV/c 2 

Bubble chamber: 
(freon filled) [10] 0.9±g;|f GeV/c 2 

(propane) [11] 0.7±0.2 GeV/c 2 

b) at Argonne National Laboratory: 

Spark chamber exp. [12] 1.05±0.2 GeV/c 2 . 

The errors quoted here are largely due to nuclear effects which make 
it difficult both to select genuine elastic events and also to determine precisely 
the relevant kinematical parameters. In fact, reaction (25) takes place on 
neutron target, namely inside nuclei. Also using propane, i.e., carbon targets 
which are comparatively small nuclei, nuclear effects such as scattering of 
protons, Fermi motion, pion absorption, cannot be neglected. The size of 
such effects has been estimated using simple models by Montecarlo meth- 
ods [13] and also by more sophisticated nuclear models [14]. Two typical 
experimental distributions of dajdq 2 (integrated over the neutrino spectrum) 
are shown in Fig. 1, 2 and 3. 

If the cross-sections dafdq 2 is measured for processes (26) as well, the 
axial form factor can be directly computed. 

In fact, from (27) and (28) 

< 30 > (SI." (a?L = ^ w, (4M£ '" ™ + A "^ 2 > • 

Experiments with antineutrinos have been attempted but, so far, have 
not yielded substantial results. In fact it is much more difficult to produce 
a clean beam due to the unfavorable ratio tz~/-k + in meson production processes 
by protons. Moreover, the process (26) produces only one charged particle 
in the final state, i.e. a [i + or an e+. No kinematical fit can be made to 
establish the nature of the event which has to be assumed a priori. 

A less direct way of measuring F A {q 2 ) is given by the neutrino inelastic 
events 

(31) v-j-Jf ->r+jr+7r, 



122 



C. Franzinetti and G. Stoppini 




q 2 CGeV/c) 2 



Fig. 1. - The q 2 distribution of neutrino events observed in the CERN freon- 
filled heavy liquid bubble chamber (HLBC). The different curves refer to 
different axial-vector form factor assumed [11]. 



where JSP, JV" indicate a nucleon, either a proton or a neutron. The hadronic 
current is, in this case, given by a much more complicated expression than (1 1). 
It can be seen that the most general matrix element is formed by 8 vectors 
and 8 pseudo vectors. Conservation of the vector current reduces the number 
of terms from 16 to 14 and the « isotriplet vector current hypothesis » deter- 
mines the vector current directly from electro- and photoproduction data. 
The axial-vector part can be calculated with the help of dispersion relation 
technique. Let Mi be the projection of the axial-vector amplitude on the 



Axial-vector coupling 



123 



20" 



Experimenhal values 
rr conlTibuMon 



to 

c 

> 

<D 
O 

d 



10- 








TheoreMcal curve 
for M A = 0.7GeV/c 2 







0.2 0.4 0.6 



0*8 



1.0 



q 2 CGeV/c) 



Fig. 2. - The q 2 distribution of neutrino events observed in the CERN propane- 
filled HLBC. The open circles (o) indicate the estimate contribution to the 
background of nonelastic events of the type v+n -> [I'+p+iz . The filled cir- 
cles (•) indicate the experimental values. 



/-th multipole and let us assume that Mi satisfies the dispersion relation 



(32) 



Mi(W) = Mf + 



7tJ 



Im Mj(W)dW r 
W'—W 



where Mf is the contribution of the Born terms to the same multipole. 

The solution of eq. (32) is proportional to Mf and thus is a linear function 
of F A (q 2 ). Thus (*) an analysis of « single pion » events gives in the end an 
estimate of F A (q 2 ). 

This analysis is much more elaborate and perhaps less valid than that 
on the elastic events, due to the various assumptions and approximations 



(*) The induced pseudoscalar term, which is proportional to the lepton mass, does 
not contribute appreciably to the axial vector matrix element. 



124 



C. Franzinetti and G. Stoppini 



-a 
a> 



o 
o 



50- 



40 




elastic events 
+ inelastic bkg. 

M A = 0.84GeVl e(ashc 
M A =0.5 GeVjevents 



1.2 14 CGeV/cV 



Fig. 3. - The q 2 distributions of neutrino events observed in the CERN spark 

chamber [9] and selected according to the criterion: E v ~ 1.4 GeV; cos cp en 0.8. 

The curves give the theoretical distributions estimated for the elastic events + 

+ inelastic background for different values of M A . 



which are involved in the calculations. However, if it is assumed to be valid 
and the theoretical curves thus obtained are fitted on the experimental distri- 
butions, one obtains (assuming F A (q 2 ) to be as in eq. (29)): 

a) experiment using the C 3 H 9 -filled bubble chamber: 

for the reactions v+p -^(jt+^ + +P, M A = 1.250±0.350 GeV/c 2 
for the reactions v+J^->[x-+7u +JV", M A = 0.850±0.250 GeV/c 2 

b) experiment using the freon-filled bubble chamber 

for the reaction v+ JV -> {x~+ tt + JY", M A = 0.900±0.250 GeV/c 2 , 

where JV or JV" indicate a nucleon, either a proton or a neutron. Thus within 
these large limits of uncertainty it appears that the « axial-vector radius » 
of the nucleon does not differ appreciably from the vector form factor. 

Theoretical predictions on the form of F A {q 2 ) are rather vague at present. 
Sum rules, connecting the nucleon form factors (and hence F A {q 2 )) to the 
inelastic structure factors have been obtained by Adler [5]. « Structure fac- 
tors » are the quantities oc(q 2 , W), ft(q 2 , W), y(q 2 , W) in the expression of the 
cross-section 



& 2 o 



dQ t d£ t 



(v+^ -> C-+H) = G ( ^f ° Ei [g 2 a + 2£ v £ t cos 2 1 fi - 



■(Z, + Et)q*y\ 



Axial-vector coupling 125 

where t indicates a lepton, q 2 = Q? v — pt) 2 , £ v the neutrino energy, and <p 
the angle of emission of the charged lepton with respect to the neutrino; 
W is the mass of the final hadronic system H, 6c=0 A f^6 w . However a, /?, y 
are far less known than F A (q 2 ) and also less easy to be determined over a 
wide range of q 2 and W. Thus these rules — which are in fact tests of local 
commutation relations — are in general of little help to determine the axial- 
vector coupling. 

4"2. Low-energy single pion electroproduction. - Symmetry considerations 
involving weak and electromagnetic interactions, suggest the existence of 
similarities between process (31) and the process 

(33) e+J^e+JV'-l-Tr, 

so that one has to expect to be possible to express the amplitudes for both 
processes in term of common form factors, namely the vector and axial vector 
form factors. 

Under the hypothesis of a single photon exchange between the electron 
and the hadronic system, process (33) is equivalent to a single pion photo- 
production process induced by an off-mass-shell photon so that one has to 
expect the existence of both transverse and longitudinal amplitudes [15]. 
Furthermore, for given initial and final electron four-momenta, the e.m. 
radiation possesses a well-defined polarization state described by the polariza- 
tion parameter 

-l 



(34) 



l+2^- 2 tg^' 
^ k 2 8 2 



which measures the transverse linear polarization of the virtual photon. 
Here k^ is the photon four-momentum and \p the laboratory electron scat- 
tering angle. 

The differential cross-section for scattering into the electron solid angle 
dQ% measured in the laboratory and into the pion solid angle dQ„ measured 
in the 7>JV cm., is given by 

(35) d ' tf - * E '\ k \\\ jTV-i^ 

y } dE'dQidQ K 2n 2 Ekl y ' dQ n ' 

where E, E' are the initial, final laboratory electron energies; \k\, k are the 
laboratory photon 3-momentum and energy (k = E — E'); daJdQ^ is the 
7C-JNP cm. differential cross-section for pion production by a virtual photon 



126 C. Franzinetti and G. Stoppini 

and can be written as 

( 36 ) U-^ = ^t + ^l^l + 5cos2?> + Ccos9?, 

where \q\ is the pion 3-momentum ; <p is the angle between the planes of 
initial and final electrons and initial electron and final pion; <£l = (k z jkfy£ > . 

The first term represents the cross-section for pion production by an 
unpolarized, transverse virtual photon; the second term is the cross-section 
for pion production by a longitudinal photon; the third term arises from 
the interference between transverse states while the fourth from interference 
between longitudinal and transverse states. 

That is all can be inferred by the hypothesis of a single photon exchange 
and by the use of the properties of the electromagnetic field. An evaluation 
of the At, A l , B, C coefficients implies dynamical considerations on the in- 
teraction of the e.m. field with the hadronic system and we have to expect, 
on general grounds, that they will depend on k 2 , q 2 , and 6 n (the 7u-J\P cm. 
angle between photon and pion). If eq. (35) is integrated over the 6 and <p 
variables one obtains 

(37) afe = £• §' 13 (1 " gyl M * 2 ' w) + '"**> w)] • 

where 

o o 

j-1 or T = n \A T d(cos 0„) , j-^. er L = 2n \A L d(cos &„) , 

1^1 J \9\ J 

W = \/ml + |?| 2 + VM 2 + |^| 2 , 

and or, a^ measure the total absorption cross-section for transverse and, 
respectively, longitudinal virtual photons. In the following (37) will be referred 
as « total cross-section ». 

By using the PCAC and current algebra hypotheses, it is possible to 
obtain definite predictions on the single pion electroproduction amplitude for 
the pion four-momentum q^^O. Under this condition, that for external 
pions implies w Jt ->0, Adler and Gilman [16] and Riazuddin and Lee [17], 
have been able to obtain sum rules through the comparison of a standard 
dispersion calculation with the PCAC-current algebra approach. 

In general, an evaluation of a zero pion mass amplitude runs through the 
following steps [8]: 



Axial-vector coupling 127 

a) An off-mass-shell amplitude is defined and by using the PCAC 
hypothesis a reduction formula is obtained as a sum of two terms: for pro- 
cess (33) one term contains the fourth-component of the axial current-vector 
current commutator (equal-time commutator) and the other contains the 
axial current-vector current time-ordered product. 

b) Current algebra commutators can then be used to obtain the term 
due to the equal-time commutator while, when q^ -» 0, in the time-ordered 
product, only the single nucleon pole survives to which continuum contribu- 
tions beginning at q 2 = — 9m 2 . have to be added. For elastic iz-Jf scattering, 
for instance, in the ^-complex plane, the single nucleon pole is at Re q = 
= — m\l2M while the threshold unitarity cut starts at Re q = m n . When 
m n -> both the single nucleon pole and the threshold go toward q = 
so that one can expect that the amplitudes evaluated for q n -> (including 
only the single-nucleon pole contribution) represent a good approximation 
to the low energy physical amplitudes. It has to be noticed that the pole 
terms obtained in the limit q n -^0 do not contain the pion pole. 

c) An estimate of the low energy amplitude on the mass shell is then 
performed extrapolating from the zero-pion-mass expression. 

The first step, as a definition, implies a certain amount of arbitrariness. 

The problem of the extrapolation of the zero-pion-mass result to physical 
pions is not trivial and the procedure to solve it is not unique. In general, 
the used procedures are all based on the assumption that the off-mass-shell 
amplitude is a smooth function of q as would be expected in a perturbation 
expansion, based on a Lagrangian field theory for which the PCAC hypothesis 
holds. 

We quote here two extrapolation methods which allow to obtain def- 
inite predictions for process (33): 

Method I. Balachandran et al. [18], following a proposal of Sugawara [19] 
and Suzuki [20], obtain local statements about physical quantities from cur- 
rent algebra and are able to approximate the physical amplitude in terms 
of the value of the off-mass-shell amplitude and its derivatives evaluated at 
an appropriate unphysical point. 

Method II. To perform the extrapolation to physical pions, Fubini and 
Furlan [21] propose the use of mass dispersion relations and give definite 
prescriptions on the path along which it is convenient to extrapolate, namely 
a path along which the amplitudes are almost constant. The Fubini-Furlan 
method gives predictions for physical amplitudes in defined points of the 
physical region: for process (33) this is done at the nucleon Breit-threshold. 

Method I predicts the pion-nucleon scattering length with a precision 



128 



C. Franzinetti and G. Stoppini 



of ~ 10 %. Method II predicts the same physical magnitudes with a precision 
of 10 to 20 %, while applied to threshold single pion photoproduction predicts 
the 71+ threshold matrix element to better than 10% and remarkably well 
the threshold tz~/'k + ratio. 

Method I has been used by Gleeson et al. [22] to evaluate the positive 
pion electroproduction amplitude at the physical threshold. At the physical 
threshold, we have 



w 

l\9\ 



T TT+n 
7 T 



r 7T+n 



«-0 



«-0 



= \E&*(k\q = 0)\ 2 
= \Lt + \k\q = Q»\\ 



-£o+, L 0+ being the transverse and longitudinal electric dipole J =\ transition 
amplitudes. 

The Gleeson et al. predictions are 



Ef + \k\ q = 0) = V2& [" n GXiW^F^) - F A (k*)] ( 1 + ^ J 



m\k\ q = 0) = <flQ \ l + 4tfi G"p-/0 



where 



([ 



k* 



G?(k*)-^F^ 



i(* 2 )( 



+ 



-Ai^)-2F A (^) 



k 2 \ 



kF x {k 2 ) = F\(k 2 ) — Fl{k 2 ) 

Q _k^ V 2 (4M 2 + A: 2 )* 
~c2M gA 2M 2 + k 2 



G\{k 2 ) = Fl{k 2 )- 



4M' 



F$(k 2 ) 



and c is denned through the PCAC relation d^A* = ^/2~\c + \(p+. The authors 
claim that, within the method, the amplitudes are correct to better than 15 %. 

Method II has been used by Furlan et al. [23] to evaluate the single tt + elec- 
troproduction amplitude at the nucleon Breit-threshold. For & 2 < 10 4 fm 
the \q\ values corresponding to the nucleon Breit-threshold range from ~ 10 
to ~ 35 MeV/c so this prediction can be easily used, by only introducing 
kinematical factors, to evaluate the physical threshold amplitude. Their 
result depends on F A and F , but using the relation between them given by 



Axial-vector coupling 129 

the pion pole dominance hypothesis it can be written as 



E ^ k2 >« = °^w{k 



\Fa(<) + ^G%\t)G A (0) 



+ <5 



where F A (t) is the axial- vector form factor of the nucleon; G M {t) — F^ + F% 
is the Sachs nucleon form factor; t = k 2 — m\ — 2m n k is the nucleon four- 
momentum transfer at threshold; d and y are corrections to the main soft 
pion term and the authors are able to give a precise recipe to evaluate them. 

Both the above formulas, at the limit A: 2 -»0, reproduce the features 
of the Kroll-Rudermann theorem. 

For the reaction e+p-^7r + +n-f e, Amaldi et al. [24] have recently pub- 
lished an experimental result of the threshold amplitude at k 2 = 5frrr 2 . The 
experiment was a measurement of total cross-sections in the interval of \q\ 
from 30 to 80 MeV/c. The experimental apparatus is shown in Fig. 4. It 
consists of an electron magnetic channel through which k 2 and q are deter- 
mined for fixed incident energy. By detecting the coincidences of electrons 
with the protons of the concomitant processes 

/ T + e +P 
e+P x 

x 7i°-fe+p 

it is possible to evaluate what fraction of the single electron arm rate is attri- 
butable to process (33). As a result of this subtraction method, the authors 
obtain the data of Fig. 5 already corrected for radiative corrections. By fitting 
the data with a fourth-order \q\ polinomial, the authors obtain, at 5fm -2 

lim Iri JTTX^l = ( 4 - 9 ± °- 7 ) x 10 ~ 31 JflL, , 
|«|-*o [\q\ dQidE'] sr(GeV) 2 /c 

which is proportional to \Efc n (k 2 , \q\ = 0)| 2 +<^ L |Ls; n (& 2 , \q\ =0)| 2 . A com- 
parison of this experimental result with the Gleeson et al. predictions where 
use is made of the axial form factor parametric representation 

A: 2 \- 2 



0+5) 



gives 

M A = (1.03 ± 0.07) GeV. 



130 C. Franzinetti and G. Stoppini 

4 PROTON TELESCOPE 




ELECTRON MAGNETIC 
CHANNEL 



^frA --' 



1m 



HZHE 2 

Fig. 4. - Experimental set-up for tt+ electroproduction total cross-section meas- 
urements: channel 2 of proton telescope detects mainly tt° protons while chan- 
nel 1 detects wide-angle bremsstrahlung protons. 



The fit made by using the Furlan et al. amplitude gives a similar value 
of M A . 

To decide the most appropriate extrapolation procedure, a comparison 
of the predictions with the threshold electroproduction amplitude is more 
efficient than with low energy pion-nucleon scattering and threshold single 
pion photoproduction. As a matter of fact, the electroproduction physical 
threshold amplitude is a function of the virtual photon mass and a more 
complete experimental investigation of the threshold region could indicate 
the best representation of the amplitude and, in the mean time, give an 
efficient way to measure F A (k 2 ). The extension of the results to a range 
of k 2 is now in progress by using an electron-neutron coincidence method. 



Axial-vector coupling 



131 



<G"> 



m 



20 



15" 



OJ 

E 

u 
OJ 



10 



e+p-».e+n+TT + 



• E = 800MeV 
o E = 780 MeV 



0.2 



04 



0.6 



0.8 






Fig. 5. - Experimental results on tc+ electroproduction (e+p -> e+n+^+) near 

threshold referring to two different settings of the primary electron energy. 

(•): E= 800 MeV; (o): E = 780 MeV. 



Furthermore, it would be useful to allow a still more detailed comparison 
with the predictions, obtaining separate experimental informations on the 
threshold transverse and longitudinal amplitudes. This could be accom- 
plished by performing measurements, at the same k 2 value, for different 
values of $ but unfortunately this method encounters serious counting rate 
difficulties. Another method could consist in measuring the cm. angular 
distributions of pions for low values of \q\. In fact, for fixed k 2 , \q\ and 
d n , the <f> distributions gives A = At + ^l^l, B, and C separately. Limit- 
ing ourselves to s and p waves we obtain for A, B, and C expressions of 
the type 

|tf|~M = A + A x cos 6 n + A 2 cos 2 B n 
Iql^B = smd n (B + B lC osd n ) 
I^C^Cosin 2 ^, 

where A , ... are only functions of k 2 and \q\ 2 . At threshold only A is dif- 
ferent from zero. Of the other coefficients can be measured the slopes at 



132 C. Franzinetti and G. Stoppini 

threshold. Having obtained these experimental informations, one can see by 
using a multipole expansion that it is possible, within reasonable hypotheses, 
to separate the threshold longitudinal and transverse amplitudes. An exper- 
iment is now in progress at NINA (Daresbury) to measure low energy an- 
gular distributions of the pion. 



REFERENCES 



[1] N. Brene, L. Veje, M. Roos and C. Gronstron: Phys. Rev., 149, 1288 (1966). 
[2] F. Eisele, R. Engelmann, H. Filthuth, W. Fohlisch, V. Hepp, E. Leitner, W. Pres- 

ser, H. Schneider and G. Zech: Z. Phys., to be published. 
[3] C. S. Wu and S. A. Moszkowski: Beta decay (Wiley Interscience, New York 1966). 
[4] Y. Nambu: Phys. Rev. Lett., 4, 380 (1960); M. Gell-Mann and M. Levy: Nuovo 
Cimento, 16, 705 (1960); J. Bernstein, S. Fubini, M. Gell-Mann and W. Thirring: 
Nuovo Cimento, 17, 757 (1960). 
[5] S. L. Adler: Phys. Rev., 140, B 736 (1965). 
[6] W. I. Weisberger: Phys. Rev., 143, 1302 (1966). 
[7] S. Fubini and G. Furlan: Physics, 1, 229 (1965). 
[8] S. Weinberg: Phys. Rev. Lett., 17, 616 (1966). 

[9] M. Holder: Thesis Aachen Interner Bericht, 19 (1967); M. Holder, A. Stande, 
A. Bohm, H. Faissner, H. J. Steiner, J. K. Bienlein, G. von Dardel, F. Ferrero, 
J. M. Gaillard, H. J. Gerber, V. Kaftanov, F. Krienen, C. Manfredotti, M. Rein- 
harz and R. A. Salmeron: Nuovo Cimento, 57 A, 338 (1968). 
[10] C. Franzinetti and E. Young: CERN, NPA/Int. 66-19. 

[11] Yu. Budagov, D. C. Cundy, C. Franzinetti, W. B. Fretter, H. W. K. Hopkins, 
C. Manfredotti, G. Myatt, F. A. Mezrick, M. Nikolic, T. B. Novey, R. B. Pal- 
mer, J. B. M. Pattison, D. H. Perkins, C. A. Ramm, B. Roe, R. Stump, W. Venus, 
H. W. Wachsmuth and H. Yoshiki: in press. 
[12] R. L. Kuston, D. E. Lundquist, J. B. Novey, A. Yokosawa and F. Chilton: Phys. 

Rev. Lett., 22, 1014 (1969). 
[13] C. Franzinetti and C. Manfredotti: CERN NPA/Int. 67-30. 
[14] A. Orkin-Leconrtois and C. A. Piketty: Nuovo Cimento, 50 A, 927 (1967). 
[15] N. Dombey: Rev. Mod. Phys., 41, 236 (1969). 
[16] S. L. Adler and F. J. Gilman: Phys. Rev., 152, 1460 (1966). 
[17] Riazuddin and B. W. Lee: Phys. Rev., 146, 1202 (1966). 
[18] A. P. Balachandran, M. G. Gundzik and F. Nicodemi: Lectures on Theor. Phys., 

9B, 361 (1967). 
[19] H. Sugawara: Phys. Rev. Lett., 15, 870 (1965). 
[20] M. Suzuki: Phys. Rev. Lett., 15, 986 (1965). 
[21] S. Fubini and G. Furlan: Ann. Phys., 48, 322 (1968). 

[22] A. M. Gleeson, M. G. Gundzik and J. C. Kuriyan: Phys. Rev., 173, 1708 (1968). 
[23] G. Furlan, N. Paver and C. Verzegnassi: Nuovo Cimento, 62 A, 519 (1969). 
[24] E. Amaldi, M. Balla, B. Borgia, G. V. Di Giorgio, A. Giazotto, P. Pistilli, 
S. Serbassi and G. Stoppini: to be published on Nuovo Cimento. 



Old Problems and New Ideas 
in Elementary Particle Physics. 

S. FUBINI 

Istituto di Fisica - Universitd di Torino 



After twenty years of elementary particle physics we are still very far 
from having a satisfactory theory accounting for the different phenomena in 
this field. 

Although one might feel that this situation is somewhat justified by our 
experimental knowledge of the spectrum and of the interactions of elementary 
particles, I think that the main reason is indeed theoretical. 

In our field we are dealing with an extremely relativistic problem, where 
the binding energies are of the same order of the masses, so that creation and 
destruction of particles is the most usual phenomenon. This has not as yet 
allowed us (and probably never will) to isolate from the general structure 
some less complicated system for which to construct a simple self-consistent 
theory. 

As an example we can consider a two-body process: 

(I) A+B->C+D. 

The relativistic nature of the problem requires that the same amplitude 
represents at the same time the « crossed» processes 

(II) A+C^B+D 
and 

(HI) A+D->B+C, 

where A, B, C, D, are the antiparticles of A, B, C and D. 

We denote by Pa, Pb, Pc, Pd the four momenta of the particles and define 



134 S. Fubini 

the invariant quantities: 

(1) 



s =(pa + Pb) 2 
t = (Pa—Pc) 2 
u = (pa—Pd) 2 



which represent the square of the cm. energies in channels (1), (II), and (III), 
respectively. 

We recall the kinematical relation 



(2) 



s + t + u = 2 M\ 



The situation can be thus represented by the triangular plot of Fig. 1, 
in which s, t and u are represented by the distance of our point from the 
s, t and u axes. 

In the case of equal masses the three physical regions for processes (I), 
(II) and (III) are represented by the cross-hatched regions in the plot. Those 
regions are indeed disconnected. Since we are dealing with analytic functions 




U t 

Fig. 1. - See text for explanation of symbols. 



Old problems and new ideas in elementary particle physics 135 

of s, t, and u, continuation between one physical region and another is possible- 
One of the fundamental problems in elementary particle physics is to con- 
struct a scattering amplitude which can be analytically continued from one 
region to the other and satisfies all fundamental physical constraints (like 
unitarity) in all regions. 

The triangular plot in Fig. 1 provides us with a simple way of recognizing 
how much of relativistic dynamics is present in one problem. This is achieved 
by comparing the size of the fundamental (s, t, u) triangle with some char- 
acteristic energy of the problem, for example the average distance (or better 
Ml— Ml) between energy levels. 

In questions of low energy nuclear physics the size of the triangle is of 
the order of the square of the mass of the nucleus, whereas the average level 
spacing is of the order of mega electronvolt. In this case one can live 
happily without worrying about the existence of crossed channels. On the 
other extreme, for pion-pion scattering, the size of the triangle (~4m%) is 
of the same order (and even smaller) as compared to m* — m\ . We thus 
face the much more difficult problem of providing an amplitude which is 
simultaneously « reasonable » in all three channels. 

In particular, if we want to discuss a model in which the main effect is 
due to the exchange of resonant states in all channels we have to look for a 
crossing invariant generalization of the Breit-Wigner formula. 

During the last few years a new point of view has been developed in this 
respect. This has led to the so-called « duality principle » which (in the case 
of two-body collisions) requires that the sum of all resonant contributions in 
the s channel does automatically represent the sum of contributions to the 
crossed t channel. 

A simple, beautiful realization of duality is given by the Veneziano rep- 
resentation in which the two-body scattering amplitude has the form 

(3) A(s, t, u) = Ax(s, t) + Au(t, u) + A m (w, s) , 

where 

i 

(4) Afat) = fx-*<«>-i (1 - x)-«(0-id;c 

o 

and analogous expressions for An and Am. 

The exponent oc(s) appearing in eq. (4) is a linear Regge trajectory 

(5) x(s) = as + b . 



136 S. Fubini 

Each integer intercept oc(s) = n corresponds to the mass of one (or more!) 
resonance. Equation (5) represents the case in which all resonances are taken 
in the limit of zero width. 

Let us go back to eq. (4); substituting x= \—y we obtain for A\(s,i) 
the reciprocal form: 



(6) 



Aifo = f(l — y)-«(s)-ly-«(t)-l dy # 



If we now expand the term (1 — x) -< *^ )-1 (in eq. (4)) and in powers 
of x we obtain: 

(7) A I (s,t) = 2 Cnit) 



o oc(s) — n 



Ai(s, t) is thus written as an infinite sum on s channel resonances. 

On the other hand, if we expand the term (1 — _y)-*(*)-i (in eq. (6)) in 
powers of y, we obtain for Ai(s) the completely equivalent form 

(8) A,(M) = 1 C * (5) 



Va(0 — n 



as an infinite sum of t channel resonances. 

Besides the perfect duality property exhibited in eqs. (7) and (8) the Vene- 
ziano formula has many appealing features. The asymptotic behaviour follows 
the Regge law in all three channels; moreover all constraints due to super- 
convergence and finite energy sum rules are automatically satisfied. 

A recent wonderful development has been the generalization of the ex- 
pression for the two-body amplitude to processes with any number of external 
lines. Those amplitudes do again satisfy duality (which is now a much more 
stringent requirement because of the larger number of crossed channels) 
and exhibit the well-known multi-Regge behaviour in all channels. 

All this might suggest that we are dealing, not only with a beautiful model 
for scattering amplitudes but, maybe, with the starting point of a new general 
theoretical scheme for elementary particle physics. Time will tell. At present 
the weakest point is the absence of unitarity. The resonant dual amplitudes 
look very much like generalized Born approximations. Until now, attempts 
of introducing unitarity in a systematic way have met with great difficulties. 

Waiting for some new ideas which might get us out of this deadlock; 
some interest has been devoted to the more modest question of understanding 
the nature of the resonant states appearing in the dual models. It has been 



Old problems and new ideas in elementary particle physics 137 

found that the different intercepts: 

(9) a(s) = n 

do not correspond to single resonances, but to very degenerate states. For 
large n the degree of degeneracy increases as exp [en], i.e., exp[c'\/5]- 
It appears that this rapid growth of the number of levels with energy is 
needed in order to satisfy all constraints related to duality. 

This somewhat unexpected feature of the level structure of dual resonant 
models is by no means unreasonable. It tells us that the concept of a single 
resonances is a useful one only at sufficiently low energies (in the giga electron- 
volt range). At larger energies the number of levels for energy intervals 
becomes so great that we shall practically have to deal with a continuum in 
which the single levels will lose their individuality. 

Thus we can use the dual resonant models as starting points of statistical 
considerations about average properties of such levels. 

Let me close this short survey by an optimistic note. Although the still 
unsolved questions look formidable and many years of hard work might 
still be needed, I think that the new approach based on duality is a modest 
step in the right direction. This may finally lead to a satisfactory theoretical 
treatment of strongly interacting elementary particles. 



High-Energy e + -e~ Annihilation into Hadrons. 

R. Gatto 

Istituto di Fisica delVUniversita - Padova 
Istituto Nazionale di Fisica Nucleare - Sezione di Padova 



1. - In 1961 the Italian Institute for Nuclear Physics, under the Presidency 
of Amaldi, took the important decision to start the construction of Adone, 
an e+-e~ storage ring designed to reach a total energy of 3 GeV. The machine 
has recently been successfully operated [1]. In the meantime spectacular 
developments in the field have taken place at Novosibirsky and at Orsay [2] 
thus encouraging the hope of further interesting experimental results. 

Investigation of the theoretical aspects of e + -e~ collisions had started 
very early in Rome and in Frascati [3, 4], leading to the conclusion that 
e + -e _ storage rings, when available, would be of foremost importance to 
the development of high-energy physics. In the last Section of the paper 
in ref. [4], Cabibbo and I discussed two points that have appeared to be of 
some interest recently [5-8] : a) the connection of e+-e~ cross-sections to hadronic 
contributions to vacuum polarization; b) the possible asymptotic behaviours 
of the cross-section. In this note I shall further develop such points essen- 
tially with the aims to provide a classification of cross-section behaviours 
and related sum rules, and to illustrate the possible underlying physical 
interpretation. In relation to the latter point we shall find interesting con- 
nections to the concept of compound field algebra (CFA). CFA has appeared 
of interest in an investigation of leading divergences in weak interactions 
relevant to a theory of the Cabibbo angle [9]. 

Using the Yang-Mills theory [10] and the developments by Lee, Weinberg 
and Zumino [11] to provide a frame for the discussion, we shall here sug- 
gest the classification summarized in Tables I, II and III. We hope that 
higher energy colliding beam experiments may bring decisive information in 
choosing among the alternatives presented in the Tables. 

2. - Here I shall briefly summarize some of the older results by Cabibbo 
and myself [4] which are relevant to the present discussion. 



High-energy e + -e~ annihilation into hadrons 139 

Table I. - Finite field algebra (FFA). 



/■ 



Q(o 2 ) 2 Z 

da 2 < oo m n <. oo — >0 

Z 

\da 2 Q (o 2 )<co g ^ Z>0 

4U )f<°° 

(^^(?)f i s tne Feynman propagator for the gauge particle.) 

Asymptotic behaviour: g 2 q(o 2 ) -> 0. 

a(s) vanishes more rapidly than s~ 6 (*). (For instance o(s)~s~ 6 (log s)~ 2 , etc.) 

e.m. mass differences expected to be infinite (i.e., uncalculable) ; finite Schwinger term in 

[j Ji\; finite c-number term in [8^ — dj ,j k ]. 

(*) In cm., s = IE. 



We assumed that the analysis could be reasonably limited to lowest 
electromagnetic order, at least for a finite range of energies. One calls F a 
set of hadronic final states produced according to 

(1) e++e-^F 

and gf(E) the cross-section for such a process at energy E of e+ in cm. (total 
energy in center of mass = 2E). The set F will contribute a term n F (K 2 ) 
to the absorptive part, n(K 2 ), of the photon propagator. Here K 2 is the 
virtual photon momentum 

(2) K 2 = — 4E 2 . 

It was stressed in ref. [4] that, for any set of final states F, n F and of are 
related through 

(3) a F {E) = ~n F {-4E 2 ) 

and that the existence of such a relation was indeed one of the most inter- 
esting aspects of the theory of e 4 -e~ annihilation. Relation (3) is equivalent 
to the relation 

(4) q f (4E 2 ) = -^- 2 o f (E), 



between a F {E) and the contribution from the set F to the spectral function 



140 R. Gat to 

Table II. - Divergent field algebra (DFA). 



da 2 <oo /w„->oo — >0 

o 2 Z n 



dCT 2 ^((T 2 ) = oo gl^oo Z->0 (Z ^0) 

^(0) F < oo 

e(ff 2 )^o 

0(i-) vanishes more rapidly than 5 -4 (for instance o(s) ~ 5 -4 (log s)~ 2 , etc.). 

e.m. mass differences infinite; finite Schwinger term in [j , y'J; infinite c-number term in 



^(cr 2 ) which appears in the Lehman-Kallen representation [12] for the vacuum 
expectation value of the e.m. current commutator 

CO 

(5) <[f™ix), yr-(0)]>o = ijda*<>(cr*) (&„— ^ 3, W a*) . 

o 

We also pointed out how different assumptions on the finiteness or lack of 
finiteness of relevant integrals involving ti(K 2 ) (or equivalently, involving the 
spectral function q) would lead to statements on the asymptotic behaviour 
of a(E). In particular, some observable effects are known to depend on 
integrals 

(6) J%^ d «- 

If one wants them to be finite, the integral (6) must be convergent or, equiv- 
alently, 

(60 J«£?d*»<oo, 

or, in terms of a{E) 

(6") (dEE^(j(E) < oo . 

Equation (6") implies a decreasing cross-section, a very weak statement 
presumably. (Under such conditions the hadronic vacuum polarization cor- 
rections to g — 2 of the electron or of the muon, for instance, are finite.) 



High-energy e+-e annihilation into hadrons 141 

Table III. - Compound field algebra (CFA). 



Z 

oo — -^0 



oo Z^O 



J da 2 Q (a 2 ) = oo J ^ J w 2 

We consider what we call the « standard realization » of CFA: m„ -> oo and g -> constant. 

In this case I dcx 2 — — ~ ml and J da 2 g(a 2 ) ~ wj 

g((T 2 )~'CT 2 asymptotically: o(s)~l/s 2 . 

e.m. mass differences finite: infinite Schwinger term in [/„,/,]; infinite c-number term in 

A more stringent statement was derived in ref. [4] from the assumption that 

00 

(7) M Ua, 



be finite. Such an assumption is connected to the possible finiteness of the 
hadronic contributions to charge renormalization. From eqs. (3) and (4), 
one would obtain 

(70 j^-Woo, 

or equivalently 

CO 

(7") (dEEa(E)<oo. 

The cross-section in this case would have to decrease faster than E~ 2 , say, 
~£~ 2 (log£)- 2 , etc. 

3. - The spectral representation of eq. (5), taken at equal times, gives 
directly 

00 

(8) S(xo)<[ti m <x)J e riO)]\ = (M,< + ^,«W(*)J^e(* a ) • 



Equation (8) is one of a class of sum rules [13] that have been intensively 
exploited during last years. Equation (8) can be taken as a proof for the 



142 R. Gatto 

existence of Schwinger terms in the commutator between a space component 
and the time component of a local current. The conservation of the current 
is irrelevant to such a proof. For a conserved current, Schwinger's original 
argument [14], makes the argument quite transparent: The limit 

(9) Jim < U(0), [H, j (x)]] > <5(x ) , 

vanishes only if j o (x)\0y = 0, i.e., for a vanishing current; for a conserved 
current the limit is 

(io) iimaKU(0),y*(*)]><A*o) 

and its nonvanishing contradicts the naive calculation of the commutator 
with currents taken as bilinear forms of fields at the same space-time point. 
One solution is to redefine the currents by introducing an infinitesimal space- 
like separation e^ between the arguments of the fields, to formally compute 
the commutators, and to let s^ -> isotropically (e-limit procedure). This 
procedure (which may still be deceptive because it is based on the unrenor- 
malized fields; see, however, our discussion later) suggests a quadratically 
divergent vacuum expectation value for the Schwinger term 

(ii) «w<[/oW,y«(o)]> ~^^). 

We also recall that in quantum field theory Schwinger terms are seen to be 
related to the so-called seagulls [15]. Such a relation follows here directly 
from gauge invariance. 

4. - Sum rules for higher moments of the spectral function 

(12) (doW N Q (a 2 ), 

can formally be obtained in terms of multiple commutators 

(13) <[[...[[e,(0), P a ] P p ] ... P a ]W)]>o , 

where P^ is the total four-momentum operator and Q / ,(t)^ij / ,(x)d 3 x. 

Generally such sum rules will be divergent. Even so they may be of value 
in suggesting the asymptotic behaviour of £>(cr 2 ) or, equivalently, of the cross- 
sections. To illustrate the derivation consider, for instance, 

(14) C^Axo) = <[[[[G„(0), PJPp]Py\JM]> d(xo) = 

= - iW^UAW^W^ ) + ;;(0)H 3 <5W;;(0)> , (p 4 = #0 • 



High-energy e + -e annihilation into hadrons 143 

Inserting the spectral representation, in the form in eq. (5), one can write 

( 15 ) C^ yv d{x ) = — idixJd^dpid^d^dyjdoWQia 2 ) , 
which is the desired sum rule. 

5. - The spectral function q(g 2 ) can be obtained from eq. (5) 

(16) d(q) Q (-qZ)q* = 1 ± Jd** exp [- ftpc] (^^ ~|^) • 
In the cm. frame 

(17) Q{q)Q(-q 2 )q 2 d(q) = d(q)± ± J d*x exp [- i?*]<#™<*)#">-(0)> 

which, among other things, exhibits clearly the relation written in eq. (4) 
for each set of intermediate states F. 

In applying the sum rules one may note that, in the lowest order electro- 
magnetic approximation that we are adopting here, one can separately treat 
the isovector and the isoscalar terms in j% m : The sum rule in eq. (8) then 
gives, calling 2E = s, 

00 

(18) <5(x )<Un*MnO)]> = 3 * <5 W< 5i,ov >o= i^s ^(x) fds ? s 2 a(*) isov 



and a completely similar equation with « isovector » substituted by « iso- 
scalar ». In eq. (18) the Schwinger term has been called S. On the basis of 
(asymptotic) SU 3 one can try to use the relation : 3 <5 iso8 > = <S isov > or 

(19) \ds 2 s 2 [o is °v(s) — 3o isos (s)] = . 



Equation (19) suggests a persistent oscillatory character of Ao-= a isov ~3a' lsos , 
or a sharp decrease of such a difference. From the view point of saturation 
with resonances it would not be unrealistic to think of a persistent simul- 
taneous occurrence of T= 1 and T=0 vector mesons with relations among 
residue such that eq. (19) is satisfied. 

The model for the currents we had examined in Sect. 4, in terms of bilinear 
expressions in spin \ fields evidently suggests a quadratic behaviour in energy 
of the divergent integral in eq. (18). This means a(s)ccs~ 2 for large s 



144 R. Gatto 

(remember, however, we are only including one-photon exchange). The 
same result would obtain when including currents formed out of spin zero 
bosons. Al alternative argument for such a behaviour follows from the 
requisite of gauge-invariance on the vacuum polarization tensor n^)- The 
gauge invariant form is notoriously [16] 

(20) 7t MV (q) = ijtfxexp [iqxKTj„(x)MO)> -tt Mi tyt/j^ Q(° 2 ) 
and gauge invariance requires 

(21) <7„Jd 4 xexp [iqxKTj tl (x)j v (0)y o =d vi q i j -^ q(<7*) 

The left-hand side of eq. (21) can easily be calculated in the renormalized 
quantum electrodynamics of spin \ fermions and seen to be quadratically 
divergent. A behaviour q{<j 2 )ozo 2 is equivalent to a(s)ccs~ 2 (recall that 
s = 2E). The lack of covariance (besides gauge invariance) of the left-hand 
side of eq. (21) is an example of a frequent situation with divergent 
Feynman integrals. In the notation in eq. (18) higher moment sum rules 
are of the form 

CO 

(22) < [[H, fir ]? ^ov ( 0)] >0 = _1_ j ds 2 s 4 ^isov, 



CO 

(23) < [H[H, &rl 7? ov (0)] > = r ^- 2 jds*s« o( S ) is °\ 



and quite similar equations with « isoscalar » substituted for « isovector ». 
When applied to a model of bilinear currents from spin ^ fermions these 
equations appear all consistent with the above behaviour, q(g 2 ) oc <t 2 . (Of 
course the result may not hold when singular interactions are present.) 

6. - A rigorous Lagrangian scheme of vector dominance, including proper 
treatment of gauge-invariance, is that of Kroll, Lee and Zumino [17]. It rests 
on the idea of field-current identity. To illustrate the main point let us limit 
ourselves to the p-meson and its strong interaction. The Lagrangian density 
is supposed to be of the form 

(24) -\rn 2 2,Q»+2", 







= 


o, 


™ 2 ^ 







5 



High-energy e + -e~ annihilation into hadrons 145 

where <£' is invariant under g fi -^Q ft + g^d^A and a corresponding trans- 
formation on the matter fields. The (gauge-invariant) prescription is ^-^ 
->Q fl -\-(e/g)A fl inside j§?' (of course only isovector photons are included). 
The equations of motions 

(25) 

(26) 

establish the required field-current identity. One performs a wave-function 
renormalization q° = ^/Zq^ and introduces Z = (m/m ) 2 . The renormalized 
and unrenormalized sources J v and J® for the p-meson are directly related 

(27) /o = /v+(1 _ Zo) I^ = _- 2 L ±^\ 

where G^,, is the field tensor for the p-meson and g its renormalized coupling. 
(One conventionally defines the unrenormalized coupling as g = gi/ZZo 1 .) 
The important observation is that for m —»oo, the e.m. current (identical 
as we have said to the field) becomes identical to the unrenormalized current /° . 
This situation, current-current identity, had been studied by Gell-Mann and 
Zachariasen [18]. Always limiting to the above Abelian case one verifies 
how the spectral function sum rules now follow from the canonical com- 
mutators and the field equations. For instance, the sum rule in eq. (22' 
follows directly from the spectral representation 

(28) <[<?„*(*), <?,(0)]> = ijd0*Q(o*y(d x ,d f - d^dJAix, **) 
and the canonical commutator 

(29) %xJ[G H (x), <?,(<>)] = 1 dijdix) . 

The non- Abelian situation (SU 2 , SU 2 xSU 2 , SU 3 , SU 3 xSU 3 ) [11] presents a 
formal difficulty connected to the occurrence of ambiguous terms propor- 
tional to bilinear expressions in the gauge fields taken at the same space-time 
point in commutators of fields and their time derivatives. Such products are 
not well defined, nevertheless they presumably contribute a vanishing vacuum 
expectation value. [A rough argument is: {^^(x) <p v ( x )}o should be pro- 
portional to d^ on the basis of covariance, but ^(p i (x)(p i (x)') and <<Mx)<M*)> 

10 



146 R. Gatto 

have opposite sign. Furthermore in the non-Aabelian cases (escluding SU 2 ) 
some currents are not conserved. 

7. - In spite of the above remarks for the non-Abelian case we think it is 
useful to take the following attitude. We consider a general Yang-Mills 
theory and include in the discussion its limiting cases. This provides for a 
classification in terms of convergence or lack of convergence of the sum 
rules or, if one prefers, in terms of asymptotic behaviours of the e + -e~ anni- 
hilation cross-section. Alternatively one can present the classification in terms 
of limits on the bare quantities m and g . (We have already discussed one 
realization of the situation ra -^oo.) We shall briefly distinguish three cases: 

1) finite field algebra (FFA); 

2) divergent field algebra (DFA); 

3) compound field algebra (CFA). 

The three cases, 1), 2) and 3), are illustrated in Table I, II and III. 
From the point of view of the cross-sections we thus have: 

1) FFA: a(s) decreases faster than s~ 6 ; 

2) DFA: a(s) decreases faster than S -4 ; 

3) CFA: in the standard realization a(s) decreases as s~ 2 . 

Note that only in FFA and in DFA one has a finite A'^. In DFA and 
CFA mQ-^oo, i.e., the bare mass is infinite. The finiteness or lack of finite- 
ness of the spectral integrals is a direct reflection of the respective finiteness 
or lack of finiteness of the two c-number terms appearing in the commutators 

K*o) L/«o(*), 7/ji(0)] and S(x ) [d j ai (x) — d t j a0 (x), j pi (0)] . 

In the latter commutator, besides a ^-number Schwinger term tranforming 
as the component of a four-vector and thus irrelevant here, there also appears 
a ^-number (5-function contribution which transforms as a reducible tensor. 
This last term is responsible for the e.m. mass differences [19]. Finally we 
note that the standard realization of CFA gives for the spectral integrals a 
behaviour identical to that obtained from currents bilinear in spin \ fer- 
mion fields. (If one wants one can call them « quarks » considering the cur- 
rent inflation in the use of such a word.) That the limit of vanishing renormal- 
ization constants is relevant to a composite particle picture emerges also 
from a number of field- theoretic investigations [20, 21]. We note that CFA 
came of interest to us to discuss the higher weak orders in our theory of the 
Cabibbo angle [9]. An extension of these concepts along the lines of Wilson's 
approach to field theory [22] and employing approximate scale invariance 
has recently been developed [23]. 



High-energy e + -e~ annihilation into hadrons 147 



REFERENCES 

[1] F. Arnman, R. Andreani, M. Bassetti, M. Bernardini, A. Cattoni, V. Chimenti, 
G. F. Corazza, D. Fabiani, E. Ferlenghi, A. Massarotti, C. Pellegrini, M. Pla- 
cidi, M. Puglisi, F. Soso, S. Taffari, F. Tazzioli and G. Vignola: Lett. Nuovo 
Cimento, 1, 729 (1969). See also the pioneering work with Ada, by C. Bernardini, 
G. F. Corazza, G. Ghigo and B. Touschek: Nuovo Cimento, 18, 1293 (1960). 
[2] See for instance J. E. Auguston et ah: Phys. Rev. Lett., 20, 129 (1968); V. Auslan- 

der et al: Phys. Lett., 25 B, 433 (1967). 
[3] N. Cabibbo and R. Gatto: Phys. Rev. Lett., 4, 313 (1960). 
[4] N. Cabibbo and R. Gatto: Phys. Rev., 124, 1577 (1961). 
[5] J. D. Bjorken: Phys. Rev., 148, 1497 (1966). 
[6] J. Dooher: Phys. Rev. Lett., 19, 600 (1967). 

[7] J. J. Sakurai: in Proceedings of the 4th International Symposium on Electron and 
Photon Interactions at High Energies, 1969, Daresbury Nuclear Physics Laboratory, 
Daresbury, Lancashire. 
[8] M. Gourdin: Boulder Lectures (1969). 

[9] R. Gatto, G. Sartori and M. Tonin: Phys. Lett., 28 B, 128 (1968); Lett. Nuovo Ci- 
mento, 1, 399 (1969). See also: R. Gatto: Cabibbo angle and SU 2 xSU 2 breaking, 
Springer Tracts in Modern Physics, vol. 53, Springer-Verlag, Heidelberg (1970). 
[10] C. N. Yang and R. L. Mills: Phys. Rev., 96, 191 (1954). 
[11] T. D. Lee, S. Weinberg and B. Zumino: Phys. Rev. Lett., 18, 1029 (1967). 
[12] G. Kallen: Helv. Phys. Acta, 25, 417 (1952); H. Lehman: Nuovo Cimento, 11, 342 

(1954). 
[13] S. Weinberg: Phys. Rev. Lett., 18, 507 (1967). 
[14] J. Schwinger: Phys. Rev. Lett., 3, 296 (1959). 
[15] R. P. Feynman: unpublished. 

[16] V. N. Gribov, B. L. Ioffe and I. Ya. Pomeranchuck : Phys. Lett., 24 B, 554 (1967). 
[17] N. Kroll, T. D. Lee and B. Zumino: Phys. Rev., 157, 1376 (1967); T. D. Lee and 

B. Zumino: Phys. Rev., 163, 1667 (1967). 
[18] M. Gell-Mann and F. Zachariasen: Phys. Rev., Y2A, 953 (1961). 
[19] S. Ciccariello, G. Sartori and M. Tonin: Nuovo Cimento, 55 A, 847 (1968). 
[20] K. Hayashi, M. Hirayama, T. Muta, N. Seto and T. Shirafuji: Fortschritte der 
Physik, 15, 625 (1967); S. Ciccariello and M. Tonin: Nuovo Cimento, 58 A, 43 
(1968). It was conjectured by A. Salam: Nuovo Cimento, 18, 466 (1960) that the 
condition of vanishing renormalization constants is a field-theoretic formulation of 
the bootstrap principle. 
[21] The possible limits for m ^0 of the Yang-Mills theory are also of great interest: 
see R. Brandt and J. D. Bjorken: Phys. Rev., Ill, 2331 (1968) where the connection 
with the Sugawara model is also discussed. 
[22] K. G. Wilson: Phys. Rev., 179, 1499 (1969). 

[23] S. Ciccariello, R. Gatto, G. Sartori and M. Tonin: Phys. Lett., 30 B, 546 (1969); 
and Padua preprint (to be published). R. Gatto, Rivista Nuovo Cimento 1, 514, (1969). 



New Frontiers of High-Energy Physics. 

G. GlACOMELLI 

CERN - Geneva 
INFN, Sezione di Bologna - University of Bologna 



1. - Introduction. 

One of the « frontiers » of physics has been and will be the study of phe- 
nomena at always decreasing small distances. Progress in this field has been 
swift: The frontier passed from the study of molecules to that of atoms, 
then to nuclei and to elementary particles. In terms of distances, one passed 
from distances of the order of 10~ 8 to 10 -14 cm. The present frontier is the 
study of the so-called elementary particles; in particular we want to know 
if they are really elementary or composite. 

The progress has always been connected with the development of new 
accelerators of ever increasing energies. They allowed the production of 
secondary beams of higher energies and of beams of altogether new particles. 
Atomic phenomena could be studied with photon and electron beams of 
eV energies. The study of nuclear phenomena required beams in the MeV 
range and new types of accelerators: Cockroft- Walton's, Van der Graafs, 
cyclotrons, and betatrons produced the appropriate proton, electron, and 
photon beams. Another energy step and different types of machines were 
required before the new field of elementary particles could be entered ; new 
types of objects, unstable ones, were produced, which did not exist in nature. 
Synchro-cyclotrons and electron-synchrotrons produced tz mesons ; the 3 GeV 
Brookhaven cosmotron produced K-meson beams and allowed the study of 
various hyperons. At the 6 GeV Berkeley bevatron the antinucleons were 
discovered. The 28 GeV CERN Proton synchrotron (PS) and the 33 GeV 
Brookhaven AGS, together with some of the lower energy machines, brought 
into full swing the study of the resonances between various types of particles, 
and allowed the production of new objects, such as the muon neutrino. The 
high-energy frontier is at the moment represented by the 76 GeV Serpukhov 



New frontiers of high-energy physics 149 

IHEP accelerator, while colliding beam machines are opening up a com- 
pletely new field. 

The increase in energy between 1 eV to 33 GeV has brought about the 
discovery of three types of spectroscopies — the atomic, nuclear, and particle 
spectroscopies — whose study is best done with beams of appropriate energies : 
(1-H000) eV for atoms, (0.1^-10) MeV for nuclei, (0.1 -MO) GeV for par- 
ticles. The energy regions in between may be considered as transition energies 
or asymptotic regions for the preceding spectroscopy. The 76 GeV machine 
seems to cover such a type of asymptotic region for particle physics. 

The race toward higher energies will have as the next steps the 400 GeV 
proton accelerator of Batavia, and the European 300 GeV. A glimpse of 
what is going to happen at much higher energies will be offered by the CERN 
25 GeV pp ISR colliding beam machine and the 20 GeV pp Novosibirsk 
machine. 

Professor Edoardo Amaldi works on the frontier, in particular looking 
for new phenomena [1], and he has been since a long time the Chairman of 
the European Committee for Future Accelerators (ECFA). It is also because 
of his efforts that European laboratories can work right on the frontier of 
high-energy physics. 

In this note we shall discuss the recent beginning of the exploitation of 
the 76 GeV IHEP accelerator, comparing layouts and the first experimental 
results with those from CERN and BNL. Finally, we shall make some con- 
siderations and extrapolations for the future accelerators. 



2. - Accelerator beams. 

The experiments that can be performed at an accelerator depend criti- 
cally on the quality and quantity of the available secondary beams. In a PS, 
the beams may originate either from an internal target or from a target 
placed in an extracted proton beam. While the first system was used exten- 
sively in the past, the trend for the future seems to be that of switching from 
internal to external targets. The main reasons for this trend are connected 
with problems of radiation damage, which become serious when accelerators 
reach 10 13 protons per pulse (ppp); moreover, external beams offer greater 
flexibility. On the other hand, beams originating in an internal target may 
have better optical qualities, because of the smaller transverse dimensions 
of the source. 

Table I offers a comparison of the CERN and IHEP accelerators. 

The absence of long straight sections in the IHEP accelerator means that 
most of the secondary particle beams pass through the magnetic field of the 



150 G. Giacomelli 

Table I. - Some characteristics of the 28 GeV CERN Proton Synchrotron and of the 76 GeV 
IHEP Proton Synchrotron. 1972 projected values represent rough estimates. Of the two 
numbers indicating the number of beams, the first gives the actual number of beams, 
irrespective of branches, while the effective number of beams is computed using the Brook- 
haven criteria, keeping in mind compatibility of operation and number of branches: 
i) a beam with two or more branches is worth 1.5 beams; ii) compatible beams are worth 
one each, while two non-compatible ones are worth only 1. 



Machine CERN-PS IHEP 



Year 1969 1972 1969 1972 

(projected) (projected) 



Maximum energy 28 GeV 28 GeV 76 GeV 76 GeV 

Normal energy for counter experiments 19 GeV 25 GeV 70 GeV 70 GeV 

Average intensity per pulse 10 12 10 13 10 12 5 x 10 12 

Repetition rate 2.4 sec 2 sec 7 sec 7 sec 

Maximum burst length 0.5 sec 0.5 sec 1.0 sec 1.5 sec 

Total number of counter beams 12 15 3 7 

Effective number of counter beams 6 7 34 

Total number of bubble chambers 3 4 3 

Number of fast extracted proton beams 1 2 — 1-2 

Number of slow extracted proton beams 12 1 



machine. Therefore it is difficult to have more than one beam coming from any 
one internal target, and it is more complicated to get positive beams. Two 
negative beams and a neutral one, each originating from a different target, 
are in use in a single large experimental area. One of the charged beams 
(No. 2) covers the range (40-^65) GeV/c, while the second one covers the 
range (25-^40) GeV/c. Each of them has several possible branches. Beam- 
sharing among the three beams as well as with an internal facility is possible. 

CERN has two main experimental areas, one fed by internal targets and the 
other by external beams. The two counter areas cannot operate at the same 
time, while instead one or two bubble chamber can parasite for most of the 
time. Both laboratories have improvement programmes, as illustrated in 
Table I. 

A neutral beam may be made with only a collimator and a sweeping 
magnet. Instead, a charged particle beam requires a large number of quadru- 
poles and bending magnets; it is then possible to achieve beams of really 
good optical properties. For instance, the IHEP beam No. 2, which is the 
highest energy beam available at present, may yield 10 6 7r- of 40 GeV/c per 
burst with a momentum acceptance of ±10% and horizontal and vertical 
acceptances of 3 mrad each. 



Q 
< 

< 



10" 



io- 



XT 



< 

XT -mr 



10 



,-6 



A 

1 



a) 



P =70GeV/c 
P =550GeV/c 
0=4 mr 



A 



6000 



t, 




6500 
N 



7000 



10 



10" 



in 



in 



in 



^4rt-3 



^knXTh 
to 
Hto 
in 

cgl/) 



■vA 



-mio - 
nn r r 



X)" 5 h 



0* 



Tl" 



P =70 GeV/c 
P=A5GeV/c 
0=5 mr 




1500 1700 1900 2100 
N 



u 

in 
m 

va- 
in 

in 

eg 

in 



xr 



xr 

in 

in^. 
in 
«n 



K)" 



XT 



tf 



/rT 



W 



c) 

$=70GeV/c 
P=50GeV/c 
0=1Orar 



/ ~J£^. 



r~ 



r 



\i 



02 QA 0.6 08 1.0 
fi,(C,C 2 ) [at] 



fg=70GeV/c 
P=40GeV/c 
0=Omr 




02 04 0.6* 0.8 10 1.'2' 
P N (C;C ? ) [at] 



Fig. 1 . - Pressure curves (or mass spectra) at different beam momenta obtained 
with: a), b) differential Cerenkov counters, c) a combination of three threshold 
Cerenkov counters, d) a counter in the differential-threshold mode of operation. 



152 



G. Giacomelli 



3. - Particle separation at high energies. 

Typical particle ratios in a negative, high-energy (*) unseparated beam 
are: tc-(1), [x-(10- 2 ), K-(10- 2 ), e-(10- 3 ), p(10- 3 ); and in a positive beam: 
p(10), 7u+(100), ^.+(10-2), K+(10- 2 ), d(10- 2 ), e+(10- 3 ). Since one is usually 



IT 

A 



10T6- 



K" 

n 



12000 



16000 



20000 



N 



Fig. 1. - Pressure curves (or mass spectra) at different beam momenta obtained 
with: e) a combination of three Cerenkov counters in the threshold mode trig- 
gering a differential counter [2]. 



(*) In the following high energy will usually mean more than 5-10 GeV. 



New frontiers of high-energy physics 1 53 

interested in only one of them, the problem arises of removing the others. 
Unwanted particles may either be physically removed by means of separators 
or simply not counted by electronics means. 

The only types of separators available for high-energy beams are radio- 
frequency (RF) separators, which at the moment can be used only for bubble 
chamber beams, that is for beams of short time-duration. 

The electronics separation of different types of particles is usually per- 
formed by time-of-flight techniques or by means of Cerenkov counters. The 
time-of-flight technique is limited by the present capability of resolving about 
0.2 nsec. Many types of Cerenkov counters have been used for particle sepa- 
ration at high-energy [2]. The best resolutions achieved so far are of the 
order of A/? < 10~ 5 . Great care has to be exercised to achieve these resolutions: 
Threshold Cerenkov counters have to be many metres long; in differential 
counters, the Cerenkov light has to be chromatically corrected and the beam 
has to be quite parallel, so that a good performance of the Cerenkov counter 
also indicates an optically good beam. In a counter built at CERN, the 
index of refraction of the gas was measured by means of a laser-refracto- 
meter, providing an absolute measurement of the particle velocity and hence 
of the mass of the particle [2]. 

Figure 1 shows pressure curves at different momenta obtained by means 
of differential a), b), threshold c), differential-threshold d), and a combination 
of threshold and differential Cerenkov counter e). These curves show that 
present Cerenkov counter techniques allow the separation of kaons, when 
in 0.1% abundance, from pions up to 60 GeV/c (Fig. la-Id), and that a 
combination of Cerenkov counters is capable of rejection ratios of better 
than 109 (Fig. \ e ). 

4. - Particle production. 

One of the first experiments to be performed at a new accelerator is a 
measurement of the yield of different particles. These measurements are 
important both for practical purposes, as for instance the need for information 
for beam design, and on the techniques necessary to identify the particles, 
as well as to understand the physics of particle production at high energies. 

Photon production is easily measured by means of a glass total absorption 
Cerenkov counter [3] or a sandwich-counter, where scintillator plates are 
alternated with lead plates. These systems are really total absorption calori- 
meters, where the total energy lost is simply given by a pulse-height meas- 
urement. 

A study of the production of charged particles requires a system of 
Cerenkov counters as described in the previous chapter. Figure 2 shows a 



154 



G. Giacomelli 



10" 



10~ 3 



-5_ 






K7tT 



^n8 



v\ 



\ 



\ 



• o 
o 



P/TT 



\ 



♦\ 



• o 


E =70 GeV 


A 


52GeV 


7 


43 GeV 


a 


3 5 GeV 


+ 


20GeV 





19.2 GeV 



\ 



0.4 



0,6 



0.8 



P/P A 



1.0 



Fig. 2. - Particle ratios R vs. secondary beam laboratory momentum divided 
by the kinematically allowed maximum momentum of the heavier particle 
(K~ and p, respectively). The points represent the results of measurements per- 
formed at the IHEP accelerator operated at various energies [2]. Points corre- 
sponding to the same incident energy and secondary momentum correspond 
to different angles of production. The broken line represents the 19.2 GeV 
CERN data [4] for p-p collisions, which coincide with the dependence found 
by the same group for p-Al collisions. E = 70 GeV (• o), 52 GeV (a), 
13 GeV (v), 35 GeV (.), 20 GeV (+), 19.2 GeV ( ). 



New frontiers of high-energy physics 



155 



o 

» 10 " ; 
E 



oft 



10 -a 



PARTICLE PRODUCTION 

p*p,19.20GeV/c 

12.5 mrad (LAB.) 




12 
P (GeV/c) 



Fig. 3. - Laboratory spectra for n ± , K ± , p, and p produced in 19.2 GeV/c 

proton-proton collisions at 12.5 mrad laboratory angle [4]. The horizontal scale 

gives the secondary laboratory momentum, while the vertical scale gives the 

double differential cross-section in the laboratory frame. 



156 G. Giacomelli 

compilation of negative particle ratios produced when the internal beam of 
energy between 20 and 70 GeV strikes an aluminium target [2] ; the variable 
on the abscissa is r\ = p/p ma x, the secondary beam momentum divided by 
the maximum momentum the heavier particle (K _ , or p) may carry, assuming 
the usual conservation laws of charge, baryon number, and strangeness. 
When plotted vs. rj, the particle ratios change very little from 20 to 70 GeV 
for large values of 77, while they increase slightly for smaller values of r\. It is 
interesting to notice that the ratio of K to tc production remains small, even 
at energies so much larger than kaon threshold [2]. 

« Pionization », that is the production of a large number of pions, seems 
to be the dominant result of high-energy proton-proton collisions. 

The measurements of the absolute fluxes from an internal target cannot 
be very precise; moreover the theoretical analysis is complicated by the 
presence of a complex nucleus. Refined measurements require an external 
proton beam and a liquid hydrogen target [4, 5]. Figure 3 shows the results 
of a recent CERN experiment performed along these lines. 

Several production models have been proposed: They range from the 
statistical model with collective motion corrections [6], fireball mechanisms [5], 
etc., to semiempirical formulae of the types of Cocconi and Perkins [7]. 

These models give, for particle spectra, equations with a number of 
parameters to be determined experimentally. All the models are able to 
predict the pion data reasonably well, while the predictions of the kaon and 
antinucleon spectra are poorer. These models may at present be used as 
guiding lines, but are not expected to be very reliable at higher energies. 

5. - Cross-section measurements. 

The simplest cross-section measurement is the total cross-section; then 
follow, in order of complication, the elastic cross-section in the diffraction 
region, several two-body processes, etc. 

5T. Total cross-sections. - On the basis of the behaviour with energy of 
the total cross-sections, one may speak of two energy regions, the resonance 
and the high-energy regions, respectively [8]. In the first region (below 5 GeV), 
the total cross-sections are characterized by the presence of structures, most 
of which may be interpreted as resonances. In the high-energy region the 
total cross-sections are slowly varying functions of the energy and do not 
exhibit any appreciable structure. We shall be concerned with cross-section 
measurements in the second region. What is of interest here is the energy 
behaviour of the cross-sections and the relations among them. It was ex- 
pected that in the limit of very high energies, hadron collisions do not depend 



New frontiers of high-energy physics 



157 



on the nature of the interacting particles, nor on the specific mechanism of 
strong interactions. In particular: 

a) the ratios between the various total cross-sections should be governed 
by some internal symmetry; 

b) the total cross-sections of the particles belonging to the same isospin 
multiplet; and 

c) the total cross-section of particle and antiparticle should become 
equal (Pomeranchuk theorems [9]). 

More specific models predict the behaviour of the cross-sections as functions 
of energy. At present their predictions are contradictory: some models predict 
that the cross-sections decrease toward constant non-zero values [10]; others 
predict that the cross-sections go to zero as the energy goes to infinity [11]; 
finally other models predict that the cross-sections reach a minimum and then 
rise to finite asymptotic values [12]. 

Figure 4 shows a compilation of the high-energy total cross-sections [8], 
including the recent CERN-IHEP results at the 76 GeV synchrotron [13]. 



t AS 


HMORE 


60 


i 


GftLBRAIT 


H 65 

M 65 PN 


V 1-1 


NDEMBAU 


M 61 


} 


C1TR0N 


65 TIN 


t Dl 


DDCNS 


62 


i 


BUGG 


66 PN 


\ " 


VENKO 


62 


t 


FOLEY 


67 


{ "' 


OOENS 


63 


t 


ALLABY 


69 





2 3 4 50 

LABORATORY MOMENTUM (GeV/) 
Fig. 39 



Fig. 4. - A compilation of high-energy total cross-sections [8]. Data points are 
shown only for pp, pp, n~p, K~p, and K + p. The lines represent the results of 
the least squares fits of the total cross-sections above 5 GeV/c to the equation 



158 G. Giacomelli 

The total cross-sections for 7c~p, 7i-n (= -rc+p), K~p and K~n seem to 
have become energy independent in the region above 30GeV/c; the K + p 
and K + n cross-sections are already constant between 10 and 20 GeV/c, while 
the pp and pn cross-sections are still decreasing at 50 GeV/c. 

The total cross-sections on protons and neutrons have become almost 
identical, suggesting that at these energies the strong interaction cross-sections 
are almost independent of isospin. The K~p and K + p total cross-sections 
do not seem to come together as the energy increases. 

The available high-energy data may be fitted to simple empirical formulae 
of the type: 

(!) ffOlab) = cr(oo) + -j- , 

/>lab 

( 2 ) o"Oiab) = apf ab . 

The fittings to eq. (2) give worse % 2 than the fittings to eq. (1), suggesting 
that the data are in better agreement with finite asymptotic cross-sections. 
Table II gives the results of fitting the available data above 5 GeV/c to eq. (1) 
for d = 1 . Both statistical and systematic errors have been taken into ac- 
count by combining them quadratically. It is not clear how the differences 
between K~p-K + p and may be between pp-pp can be reconciled within the 
framework of the existing theories. Also: the available data do not allow 
a definite conclusion about which model predicts correctly the energy be- 
haviour. More experimental data are clearly needed. 

Table II. - Least squares fits of total cross-sections to the formula a = c + c-Jpi^ for 

P\ab> 5 GeV/c [8]. In order to obtain more satisfactory x 2 the systematic errors have 

been compounded quadratically with the statistical ones. No x 2 's are given for the K~n 

and pn, since the fits come from interpolated data. 



Total cross- 








Number 


section 




c° 


Cl 


X 2 


of points 


TT-p 




23.89 ± 0.04 


27.56 ± 0.50 


57.6 


84 


tc+p and 


n~n 


22.78 ± 0.06 


19.40 ± 0.60 


35.8 


66 


K-p 




20.18 ± 0.18 


24.63 ± 2.22 


22.1 


30 


K~n 




19.49 ± 0.31 


6.13 ± 3.08 


— 


18 


K+p 




17.44 ± 0.24 


- 0.22 ± 2.42 


31.6 


20 


K+n 




17.63 ±0.41 


0.55 ± 3.54 


— 


11 


PP 




42.44 ± 0.42 


108.3 ± 6.1 


24.8 


24 


pn 




42.48 ± 0.74 


87.6 ± 8.2 


— 


17 


PP 




38.29 ± 0.08 


15.24 ± 0.87 


27.1 


52 


pn and 


np 


36.48 ± 0.74 


27.3 ± 5.9 


21.9 


16 



New frontiers of high-energy physics 159 

The quark model prediction that the asymptotic 7rJV total cross-section 
is f of the asymptotic pJ\P cross-section, can be compared with 0.62, as obtained 
from Table II. 

5*2. Elastic cross-sections. - The elastic cross-sections are either slowly 
decreasing or remain constant as functions of energy. This means that the 
opacities, defined as the ratios of elastic to total cross-sections, are either 
constant or slowly decrease with energy. 

The elastic angular distributions may be subdivided into three regions: 

a) the very small angle region where Coulomb and nuclear scattering 
interfere ; 

b) the diffraction region proper; and 

c) the large-angle region, characterized by very small cross-sections. 

Large counter hodoscopes or large spark chamber arrays are usually 
employed for measuring the elastic differential cross-section. A simpler 
experiment was recently performed at the IHEP accelerator by bombarding, 
with the internal proton beam, a polyethylene target [14] (*). The recoiling 
proton was detected by means of solid-state detectors, which measured angle 
and range. It was thus possible to separate elastic from inelastic events in 
the \t\ range (0.01^-0.11) (GeV/c) 2 for proton incident momenta from 12 
to 70 GeV/c. In this angular region the diffraction pattern is well [represented 
by an exponential (Fig. 5): 

(3) £-«M 

where the slope b changes monotonously from 6^10 (GeV/c) -2 at 12 GeV/c 
to b = 11.5 at 70 GeV/c (Fig. 6). Thus the shrinking of the diffraction peak 
initiated at energies around 5 GeV, continues when the energy is increased. 
Only higher energies will be able to tell if the shrinking goes on or eventually 
saturates. Within the framework of the Regge-pole theory, the elastic scatter- 
ing data may yield the slope of the vacuum (Pomeranchuk) trajectory, which 
is found to be oc P = 0.40 ± 0.09 [14]. 

In another perspective the small angle elastic scattering is considered to 
arise from a diffraction mechanism, as the shadow of all the inelastic processes. 
In this framework, the simplest classical non-relativistic optical model 



(*) In the future the polyethylene target will be susbstituted with a gaseous jet of 
hydrogen at supersonic speed, and the experiment will cover the Coulomb-nuclear inter- 
ference region. 



160 



G. Giacomelli 



* 



1.0 



0.8 



lQ.6 



(0.4 



0.2 



fill 



58.1 T38 
1L=H07t0.18 



$ 



% 



0.01 



0.05 



0.10 



*( J 



Fig. 5. - The differential proton-proton elastic cross-section at a laboratory 
kinetic energy of 58.1 GeV in the 0.0K |f|< 0.11 (GeV/c) 2 range [14]. 



(opaque dis) [8], predicts that the interaction radius, given by 



(4) 



2Vb, 



grows from 1.23 to 1.34 fm. 

In more sophisticated optical models the size of the proton remains con- 
stant; the shrinking of the p-p diffraction pattern may then arise from the 
Lorentz contraction of the colliding particles in their direction of motion. 
The antishrinking of the pp elastic peak requires other hypotheses, such as 
that the total cross-sections are still decreasing and therefore one is nowhere 
near an asymptotic behaviour. 

The opinion on the large angle scattering is even more divided: some 
authors suggest that it is purely diffractive, possibly arising from different 
spatial structures inside the nucleon; other authors suggest that a statistical 
mechanism may play some role. 



New frontiers of high-energy physics 



161 



8,<oev/c)"* 
12 



10 



• -thisexpeziment 

o - n.* KMPHnnOBA M AP. HM6HA H965r) 

o -G.Brt-eeHtnletae.CERN(1965r) 
a - k.i. FoCey. w aC. Bzookhaven tt963r) 



uUS 



V 



h 



m 



I) 



.\P 



8 10 



20 30 



50 70 



Ertn&SP**' 



Fig. 6. - The coefficient 6 of eq. (3) for proton-proton elastic scattering in the 
diffraction region [0.0K |/|< 0.1 (GeV/c) 2 ] vs. laboratory kinetic energy [14]. 



5*3. Other cross-sections. - The study of charge exchange cross-sections 
is particularly important because their theoretical analysis is simple, at least 
in the context of Regge-pole theory, where only one exchanged trajectory 
explains the main features of the data. Present experimental information 
stops at 18 GeV, but experiments will soon be done at higher energies. 

Most of the available experimental results on the more sophisticated 
measurements are still not systematic, though a wealth of information is 
available. 

It is clear that one is nowhere close to an understanding of strong inter- 
actions and that more and more experiments, particularly at higher energies, 
are required. 

It may come as a surprise to learn that at the very high energies con- 
sidered in this report, one may still obtain information on nuclear properties. 
The absorption cross-sections a a measured at 20 to 50 GeV/c with incident 
K~, 7T _ , p, and d on a variety of nuclei have revealed that their dependence 
on the atomic number A is of the form [8, 13, 15] (Fig. 7): 



(5) 



a = a n A c 



162 



G. Giacomelli 



where a = 0.76, 0.75, 0.67, and 0.67, respectively for K", tc" p, and d. The 
large value of a for particles with small elementary cross-sections (K~, n~) 
may be explained qualitatively as being due to the fact that light nuclei are 



G q .10" 27 cm* 



2000 



1000 



500 



200 



100 



50 




Li Be C 



Al 



Cu 



Sn 



PbU 



Fig. 7. - Nuclear absorption cross-section for K~, it~, p at 40 GeV/c [13] and 
d at 25 GeV/c [15]. The lines represent the results of the least squares fits ac- 
cording to eq. (5); p = 40 GeV/c. 



not completely black for KT and n~. The antiprotons instead behave as if 
hitting a completely black nucleus (for which x should equal f). The d cross- 
sections, though poorly known, are very large, indicating both the large size 
of the antideuteron as well as a sensitivity to the nuclear matter density at 
the periphery of the nucleus. 



New frontiers of high-energy physics 163 

6. - Limits on new phenomena. 

The discovery of new phenomena is one of the most exciting results of 
the race towards higher and higher energies. When a new energy region be- 
comes available, a number of crude upper limits on new phenomena can 
be easily obtained as by-products of standard measurements. For instance, 
from the pressure curves of Fig. 1 one has upper limits for the production 
of negatively charged objects, in a rather ample mass region. Specific exper- 
iments are then required to refine these limits. 

Physicists have invented a number of particles which could exist but have 
not yet been found: the quarks, the intermediate vector boson, antiparticles, 
tachions, magnetic monopoles, etc. Maybe these objects do not exist; maybe 
they have large masses, so that present accelerators are not capable of pro- 
ducing them. We shall now discuss some of the recent limits at the highest 
energies. Measurements concern only differential cross-sections over specific 
ranges of energies and angles; therefore estimates of the upper limits for 
the total cross-sections are necessarily model dependent. 

6'1. The quarks. - The quarks were invented to explain the grouping of 
particles and resonances in unitary singlets, octets, and decuplets. Many 
searches were performed, employing accelerators, cosmic rays, and bulk 
matter methods. The quark detection is based on the fact that the charge 
of the quarks is fractional, for instance ±i or dzf ; therefore they ionize 
less than minimum ionizing particles (| and f , respectively) and their ap- 
parent momentum is larger than the momentum of normal particles. In 
particular, it is possible to have quarks with an apparent momentum larger 
than the momentum of the accelerator. Using these last properties, the two 
most recent experiments at the CERN and IHEP accelerators have yielded 
the limits quoted in Fig. 8 [16-18]. Although the limits have been computed 
for charges ±|, ±f the experiments are usually sensitive to charges in the 
range 0.3^0.8 times the electron charge. 

The interpretation of the limits quoted in Fig. 8 are ambiguous among 
several possibilities: 

a) quarks do not exist as physical entities; 

b) the conservation of some quantum number prohibits their existence 
as free entities; 

c) they are so massive as to be beyond the energies available at the 
present accelerators. 

Ambiguities of this type will be with us in any unsuccessful search for 
new objects. 



164 



G. Giacomelli 




234501 234501 2301 23 
QUARK MASS [GeV] 

Fig. 8. - Summary of quark production data from accelerators [18]. The total 
cross-sections are expressed per nucleon and have been calculated assuming iso- 
tropic cm. angular distributions and four-body phase space according to JY\JV -> 
-^JNTJV + QQ. The « diagonal » curves represent the statistical model pre- 
dictions. The curves A-F come from earlier experiments; curves H from [16] 
and curves G and / from [17]. 



A recent cosmic ray experiment reported indications for the existence 
of quarks of charge § [19]. A Wilson cloud chamber was triggered by a 
system of scintillation counters sensitive to large air-showers, initiated by 
extremely high-energy particles, estimated at about 10 6 GeV. A few low- 
ionizing particles were found in the core of the showers. This experiment 
seems to have been now contradicted [20]. 

6*2. Heavy objects with unit charge. - Limits on these objects are obtained 
from Cerenkov pressure curves such as those in Fig. 1 [2], or from time-of- 
flight measurements [2, 21]. The last method is particularly useful for heavy 
mass objects, especially when triggered by Cerenkov counters which veto 
light particles. A rough summary of the upper limits obtained in negative 
beams with momenta between 25 and 40 GeV/c at 70 GeV primary energy 
is the following: The production cross-section a m for masses m in the range 
m n <m<m^ is a m <l0~ 7 a n while, for m^< m< (5^-6) GeV, it is a m < 
< lO -9 ^, where a n is the pion production cross-sections. 

The antideuteron yield increased by almost an order of magnitude when 
the energy was raised from 30 to 70 GeV, and one may now work with about 
one d per minute. It is likely that much will be learned about antinuclei 



New frontiers of high-energy physics 165 

in general at future accelerators, as indicated by the first absorption cross- 
section measurements for d [15] and the succesfull finding of few events 
of 2 He [22]. 

6'3. Intermediate vector boson. - It is attractive to consider that weak 
interactions are mediated by a vector boson, the w. The smallness of the 
K$— Kg mass difference suggests that the w wass cannot be very large, while 
neutrino experiments indicate that it must be greater than about 2 GeV [23]. 

At Brookhaven, a number of experiments have investigated, without 
success, the mass region (2^-5) GeV, by looking for muons produced at 
large angles by the decaying boson [24] or by determining the intensity and 
the polarization of muons originating very near to the point of interaction 
of 28 GeV protons with uranium nuclei [25]. The last experiment quotes 
an upper limit of Ba w < 6x 10~ 36 cm 2 , where B is the branching ratio of the 
w into fi+v. 

At very high energies, the relativistic time dilatation makes the n and 
K mesons less liable to decay. Therefore, the muon contaminations in par- 
ticle beams will become smaller, and one can obtain a direct limit of the 
number of \i mesons produced directly at the target. At the Serpukhov ac- 
celerator such simple limits are at the level of 10~ 3 ^10 -4 of the pion flux 
at the same energy and angle. These limits are adequate only for excluding 
a strong production of the intermediate boson. 

6'4. Magnetic monopoles. - The possible existence of a magnetic pole 
would have some appealing aesthetical implications: 

a) it would re-establish the symmetry between electric and magnetic 
charges in Maxwell equations, in a formal way, not in a numerical 
way, since the magnetic pole strength is probably so much greater; 

b) it would provide some understanding of why the electric charge is 
quantized; and 

c) of why the photon mass is zero [1]. 

Also the experimental implications would be quite interesting: 

a) the monopole would ionize thousands of times more than a min- 
imum ionizing particle; 

b) it would be easily accelerated to thousands of GeV; 

c) it could be « stored » in some materials, and so on. 

A very complete review of monopole properties as well as of the present 
situation about its existence can be found in the review article by Amaldi [1]. 
Here we shall discuss further possibilities. 

The methods for detecting magnetic monopoles, at accelerators are usually 
based on the fact that they curve toward the poles of a magnetic field, that 



166 G. Giacomelli 

their ionizing power is larger than that of fission fragments, and that the 
ratio of Cerenkov to ionization loss is different from that of ordinary par- 
ticles [1]. A simple detector is a plastic material, placed a few centime- 
tres downstream from a target in a magnetic field: only heavy ionizing 
objects which get bent towards the poles can be detected. The fission frag- 
ments, already present in natural plastic materials, provide a calibration of 
the ionization, though they have to be removed (by heating) in order to 
reduce the background. A crude test on these lines performed at IHEP gave 
an upper limit of 10~ 38 cm 2 for monopole production. 

6' 5. Tachions. - In nature one finds massive particles that travel at speeds 
smaller than light and massless particles that travel at the speed of light. 
The question arises: is it possible to have particles which travel faster than 
light? Theoretically it is possible [26]: These objects would have an un- 
observable imaginary mass, they would gain velocity while losing energy 
and, since they travel faster than light, they should produce Cerenkov ra- 
diation also in vacuum. Finally, they would end up as transcendent tachions, 
with infinite velocity, zero energy, and a constant momentum [26]. Their 
theoretical interpretation requires that they travel backward in time, like 
antiparticles ; present theories may have problems with unitarity. 

Tachions could be produced at accelerators, very likely with velocities 
not too different from light. Actually the situation should be symmetric 
about c, the ordinary particles having speeds close to c on the low side, 
with tachions on the other side. 

Again some limits can be obtained from pressure curves such as those 
in Fig. 1, when they are continued to the left. A simple limit is given by 
the threshold counters when they are set to count electrons: There are no 
tachions to a limit of 10~ 3 of the pions. In at least one case (25 GeV/c 
secondary momentum), the differential counter was set to count « on the 
other side » up to velocities equivalent to « protons » : there were no counts 
to a limit of 10~ 6 of the pions. 

6*6. Long-lived particles. - It would be rather difficult to detect a high- 
energy neutral particle with a lifetime longer than 10~ 6 sec and a relatively 
small production cross-section [27]. The neutron is of such type, but is 
abundantly produced. In analogy with the neutron detection method we 
may speak of various detection methods for such objects: 

a) by missing-mass at production; 

b) by interaction in various target materials; and 

c) by activation and subsequent radioactivity. 



New frontiers of high-energy physics 167 

Let us consider, in particular, the last method. The delayed radioactivity 
would be of the normal type if an excited nucleus is formed ; it would con- 
sist of high-energy particles if a type of hypernucleus would have been 
formed. Such a delayed radioactivity, resulting in high-energy y-rays and 
electrons was searched for at CERN with negative results [28]. 



7. - Future perspectives. 

As already stated, the present frontier of high-energy physics is repre- 
sented by the 76 GeV machine. Broadly speaking one can anticipate that 
the accelerator will be used for: 

a) the study of resonances in production experiments; 

b) asymptotic behaviour, or better energy behaviour of cross-sections; 

c) systematic study of two body, quasi-two-body, and many-body 
processes; 

d) neutrino physics; 

e) searches for new phenomena, etc. 

The 400 GeV Batavia accelerator and the 300 GeV European one will 
probably follow the same lines. 

Altogether new possibilities were opened up by the colliding beam ma- 
chines, though their small luminosities have until now precluded all but 
the simplest measurements. The next generation of this type of accelerators 
is starting now with the successful operation of the 1.5 GeV e+e~ ADONE 
colliding beams of Frascati. Electron-positron colliding beams should, in 
principle, allow a detailed study of many electromagnetic phenomena at small 
distances, a precise investigation of the J PL = 1 boson resonances and so 
on, while the CERN 25 GeV pp and the Novosibirsk 20 GeV pp colliding 
beams should allow a first study of what is happening at much higher en- 
ergies. The program for the « first generation experiments » at the CERN-ISR 
is now taking shape. It is not too different from the program at a new high- 
energy proton-synchrotron : 

a) particle production; 

b) total and elastic p-p cross-sections; 

c) search for new phenomena, etc. 

Increasing energies mean physically larger accelerators, bigger laboratories, 
and much higher costs. Therefore it is clear that the number of super-high- 
energy laboratories will be small, and that they will have an international 



168 G. Giacomelli 

character. Fortunately, high-energy physics has at present no strategic im- 
plication; so it is the field of science best suited for supranational co-operation. 
Higher-energy conventional machines are becoming physically too large. 
Several solutions to this problem may be envisaged: 

a) the use of large superconducting magnetic fields; 

b) colliding beam machines; 

c) the development of completely new principles of particles accel- 
eration. 

At the same time, one has to worry about particle beam optics at these 
very high energies. Here, superconductors should be of great help in reducing 
the dimensions of the beam elements ; superconducting RF cavities may allow 
the separation of particles at much higher energies than at present. 

As far as the possibility of doing experiments at such energies is con- 
cerned, the results of the CERN-IHEP collaboration have shown that up 
to 60 GeV/c the electronic separation of particles required only established, 
though somewhat more refined, Cerenkov techniques, and did not pose any 
major technical problem. The same techniques will probably be used for 
the beams of higher energy machines. The counter beams of the future may 
incorporate many long Cerenkov counters of different types. The time-of- 
flight technique will instead remain useful when searching for heavy mass 
objects. The bubble chamber technique will still play an important role in 
first surveys of what is happening, in particular for seeing new phenomena; 
it will probably be used extensively to study neutrino interactions. Large 
magnetic spark chamber spectrometers, and eventually total absorption spec- 
trometers of large dimensions, show prospects of becoming very important 
types of detectors for the future accelerators. Cosmic rays may still allow 
a glimpse at what is happening at much higher energies: do quarks really 
exist? Will everything tend to energy-independent asymptotic limits? Will 
the Pomeranchuck theorem be violated or will something new happen, such 
as a new spectroscopy? Only accurate experiments at higher and higher 
energies will give an answer to the questions. 



I would like to express my thanks to all my colleagues [2, 13] of the first 
collaborative experiments between the European Organization for Nuclear 
Research and the Institute of High-Energy Physics, Serpukhov. Most of 
this note comes from their work and from discussions with them. I would 
also like to thank all the people who made the collaboration possible, and 
the members of the Directorate of IHEP for their hospitality. 



New frontiers of high-energy physics 169 



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ticles (Academic Press, N. Y, 1968), p. 1. 

[2] Yu. B. Bushnin, S. P. Denisov, S. V. Donskov, A. F. Dunaitsev, Yu. P. Gorin, 
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Yu. D. Prokoshkin, E. A. Razuvaev, R. S. Shuvalov, D. A. Stoyanova, J. V. Al- 

LABY, F. BlNON, A. N. DlDDENS, P. DUTEIL, G. GlACOMELLI, R. MEUNIER, J.-P. PeI- 

gneux, K. Schlupmann, M. Spighel, C. A. Stahlbrandt, J. -P. Stroot and A. M. 
Wetherell: Phys. Lett., 29 B, 48 (1969); Proc. Intern. Conf. Elementary Particles 
(Lund, 1969) and Phys. Lett., 30 B, 506 (1969). 

[3] M. Fidecaro, G. Finocchiaro, G. Gatti, G. Giacomelli, W. C. Middelkoop and 
T. Yamagata: Nuovo Cimento, 24, 73 (1962). 

[4] J. V. Allaby, F. Binon, A. N. Diddens, P. Duteil, A. Klovning, R. Meunier, 
J.-P. Peigneux, E. J. Sacharidis, K. Schlupmann, M. Spighel, J. P. Stroot, A. M. 
Thorndike and A. M. Wetherell: paper submitted to Intern. Conf. High-Energy 
Physics (Vienna, 1968). 

[5] D. G. Crabb, J. L. Day, A. D. Krisch, M. T. Lin, M. L. Marshak, J. G. Astbury, 
L. G. Ratner and A. L. Read: Phys. Rev. Lett., 21, 830 (1968). J. L. Day, N. P. 
Johnson, A. D. Krisch, M. L. Marshak, J. K. Randolph, P. Schmueser, G. J. Mar- 
mer and L. G. Ratner: to be published. 

[6] R. Hagedorn and J. Ranft: Suppl. Nuovo Cimento, 6, 169 (1968). 

[7] G. Cocconi, L. J. Koester and D. H. Perkins: Calculation of particle fluxes from pro- 
ton synchrotrons of energy 10 to 1000 GeV, in Proc. 200 GeV Summer Study (Aspen, 
Colorado, 1967). 

[8] G. Giacomelli: Total cross-section measurements, in Progress in Nuclear Physics 
(Pergamon Press, Oxford, 1970), Vol. 12, pag. 77. 

[9] I. Ia. Pomeranchuk: Sov. Phys. JETP, 3, 306 (1956); 7, 499 (1958). 
[10] V. Barger: Rev. Mod. Phys., 40, 129 (1968); V. Barger, M. Olsson and D. D. Ree- 

der: Nucl. Phys., B5, 411 (1968). 
[11] N. Cabibbo, L. Horwitz, J. J. Kokkedee and Y. Ne'eman: Nuovo Cimento, 45 A, 

275 (1966); H. Cheng and T. T. Wu: Phys. Rev. Lett., 22, 1405 (1969). 
[12] S. Frautschi and B. Margolis: Nuovo Cimento, 56 A, 1155 (1968); V. N. Gribov 

and A. A. Migdal: Yadernaya Fisika, 8, 1213 (1968). 
[13] J. V. Allaby, Yu. B. Bushnin, S. P. Denisov, A. N. Diddens, R. W. Dobinson, 
S. V. Donskov, G. Giacomelli, Yu. P. Gorin, A. Klovning, A. I. Petrukhin, 
Yu. D. Prokoshkin, R. S. Shuvalov, C. A. Stahlbrandt and D. A. Stoyanova: 
Phys. Lett., 30 B, 500 (1969) and Yadernaya Fisika (to be published). 
[14] G. G. Bzenogikh, A. Buyak, K. I. Iovchev, L. F. KiRiLLOVA, P. K. Markov, B. A. 
Morozov, V. A. Nikitin, P. V. Nomokonov, M. G. Shafranova, V. A. Sviridov, 
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1969); Phys. Lett., 30 B, 274 (1969). 
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Phys. Lett., 31 B, 230 (1970). 



170 G. Giacomelli 

[16] J. V. Allaby, G. Bianchini, A. N. Diddens, R. W. Dobinson, R. W. Hartung, 
E. Gygi, A. Klovning, D. H. Miller, E. J. Sacharidis, K. Schlupmann, F. Schnei- 
der, C. A. Stahlbrandt and A. M. Wetherell: Nuovo Cimento, to be published. 

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[24] R. Burns, G. Darnby, E. Hyman, L. M. Lederman, W. Lee, J. Rettberg and 
J. Sunderland: Phys. Rev. Lett., 15, 830 (1965). 

[25] P. J. Wandererer Jr., R. J. Stefanski, R. K. Adair, C. M. Ankenbrandt, H. Kasha, 
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[26] O. M. Bilaniuk and E. C. G. Sudarshan: Phys. Today, 22, 43 (1969). 

[27] M. Pontecorvo: 1969 Seminar at IHEP. 

[28] M. Fidecaro, G. Finocchiaro and G. Giacomelli: Nuovo Cimento, 22, 657 (1961) L. 



The K°-K° System in p-p Annihilation at Rest. 

M. GOLDHABER 

Brookhaven National Laboratory - Upton, N. Y. (*) 

C. N. Yang 

Institute for Theoretical Physics, State University of New York - Stony Brook, N. Y. (**) 



In the annihilation of stopped p by p, neutral K meson pairs are produced 
in ~10 -2 of the events. This K°-K° system is of a very special nature [1] 
and it seems worthwhile in the light of the discovery of CP nonconservation 
to discuss [2] what one can learn from a more detailed study of this system. 
The discussion is perhaps especially relevant since in the not too distant 
future it should become possible to stop a sufficient number of antiprotons 
in hydrogen in a well-defined geometry [3] to pursue statistically significant 
experiments concerning the system. 

In the reaction 

(1) p+p -> K°+K°+ neutrals , 

for fixed momenta P 1 and P 2 of the two K's, there are four states: \SS>, 
\SL}, \LS}, and \LL) where the first (or second) letter denotes the short or 
long lived nature of the K with momentum P 1 (or P 2 ). Assuming CPT in- 
variance, but not separate C, P, or T invariance, one has [4], with suitable 
normalization, 

(2) | \K s )=p\Ky + q\K), 

\ \K£>=p\K>-q\K>. 
It follows that 

(3a) \SS} + \LL) = 2p*\KK} + 2q 2 \KK} (forbidden) 

(36) \SL> + \LS} = 2p*\KK> - 2q*\RK> (forbidden) 



(*) Supported by the U. S. Atomic Energy Commission. 
(**) Partially supported by the U. S. Atomic Energy Commission under Contract 
No. AT(30-l)-3668 B. 



172 M. Goldhaber and C. N. Yang 

and 

(4a) \SS} — \LL> = 2pq\KK) + 2pq\KK} (allowed, parity = + 1) 

(4b) \LS> — \SL} = 2pq\KK> — 2pq\KK)> (allowed, parity = — 1) 

The two states (3a, b) are forbidden by strangeness conservation. The indi- 
cated parity of the two states (4a, b) is the intrinsic parity of the two K's 
taken as a single system in the final state of (1). 

1. - Experiments requiring the detection of one K°. 

It follows from (4) that in p-p annihilation one has an incoherent source 
of K s and JL L with an intensity ratio of I to 1, if one confines oneself to 
the observation of only one neutral K decay. (It seems to us difficult to ar- 
range for another source of this type.) One has thus a rather direct method 
for measuring the decay rates of K s into 7r+7r~7r°, 7i 7r°7i°, 7T"(ji+v, 7r+fjirv, 7rre+v 
or Tu+e~v. (These rates have so far not been experimentally measured. Com- 
parison of the rates of K 8 ^niv and Ki->7i;(v yields, of course, direct 
information on the AQ =£ AS question.) The sum total rate of these decays 
is expected to be ~ 10~ 3 that of K s ->7r + rc _ . The spatial distribution of the 
point of decay of K s ^'7z + n~, which is readly measurable, should be the 
same as that of any other K s decay mode. This allows for a simple method 
to separate out the K L decay modes (which form the main background events) 
from the K s decay modes under study. Scattering of the K s and K^ before 
decay can introduce uncertainties in the experiment. The amount of matter 
between the target and the detector is therefore best minimized. 

In p-p annihilation, detection of a charge asymmetry in the decay 
K L ^niv would be a direct proof of CP violation, independent of the usual 
theoretical analysis of the time dependence of the K°-K° complex, since the 
initial particles producing the K L (i.e. pp) are their own CP conjugates. 
(In contrast, the usual charge asymmetry experiments [5] examine K L pro- 
duced in hadronic collisions not involving antinuclei. They therefore demon- 
strate CP violation only if one accepts the analysis that the K L observed are 
the same whether they are produced in CP self-conjugate collisions or not. 
While this analysis is in all probability correct, there is an explicit advantage [6] 
in a direct experimental demonstration of the important phenomena of CP 
violation without having to use such an analysis.) The experiment can only 
be successful, however, if one observes — 10 7 or more K^ decays. 



The K°-K° system in p-p annihilation at rest 



173 



2. - Experiments requiring the detection of two K°'s. 

The decay amplitudes of the two states (4a) and (4b) are tabulated in 
Table I, for the case when the AQ = AS rule holds. The Table requires 
minor modifications if AQ = — AS is also allowed. The notations used in 
Table I are as follows: 

£?i = exp [— | ?l s n + /Am n] , ^ t = exp [— \l L h] , 

Am = m L —m s = 0.469 2. s , where A s = decay rate of K s , 

r}=rj + _, \rj + _\ 2 = 3.7xl0~ 6 , 
ja| 2 = 6.6xl0- 4 ; \p\ 2 oo \q\ 2 = |, 

Table I. - Decay rate into various modes at t x and t 2 . It is assumed that the AQ = AS 

rule holds (*). 



Decay 



h 


*2 


Amplitude of LS-SL 


t:\l + v 


7T+TC - 


paiSe^-n^i^) 


TTpl^V 


tt+t:- 


-qa^^+v^J?,) 


7T+TT - 


~ + 7T~ 


^(=^1^2 —^1=^2) 


TU[X + V 


TT[X+V 


^« 2 (^1^ 2 -^1^ 2 ) 


7T|X~V 


7T[i.~V 


- 9 2 «*(^1^ 2 -^1^ 2 ) 


Ufl+V 


Tr[x _ v 


paga^^^+Sr !^) 


7T+7C - 


TCfi+V 


painse^-STiSe*) 



Amplitude of SS-LL 



paiSf^-r,^^) 
qa^^ + r]^^) 

(.sr^-rfse^ 

— Ij a \Cr jt7 2 — =* 1 =* 2-) 



(*) For explanation of notation used in Table, see text. 



if 7T ev is considered instead of 7C(jlv, |a| 2 increases by a factor of 1.34. For 
those cases where the decay products at t 1 and t 2 are the same, such as 
(Tv + 7i~) ti (Tz + T~) t2 , each observed event should count as | event at t ly t 2 and \ 
event at t 2 , t x . The amplitudes are normalized such that, e.g., the probability 
of the state 2~*[|S , ,S> — \LV)\ to decay into (-k\l + v) at t x and (ti+tt - ) at f 2 * s 
\\pa(Sf x y 2 — r} r £ x Sfa^CAfdfxdfaXf). The factor (AJd^dfgXf) are common to 
all decay modes. Notice that decay into (7T[jl + v) at t 2 and (71+tt - ) at t x is a 
separate entry in the Table. 

A traditional type of experiment in which one concentrates on events 
with no final pions: 

pp^SL or SS 



174 M. Goldhaber and C. N. Yang 

has led to the conclusion that there are few, if any, events of type (4a). 
(This result is to be expected [7] if the pp annihilation occurs in the S state, 
resulting in total parity equal to — 1.) Improved statistics for this type of 
experiment would be useful. 

Observation of the interference of \LS} states with \SL} states, or of 
\SS) with \LL), in the mode (71:^)^(71+71:-)^ would lead to a measurement [8] 
of both the absolute value and the phase of rj + _ , as is obvious from Table I. 
For this experiment, the two K's could be produced together with pions. 
Even so, intensity is a severe problem, and it is doubtful that the experiment 
could be successful with less than 10 11 K pairs, or 10 13 stopped p. 

The K-K system should exhibit especially interesting properties if CPT 
invariance should be violated. Such a violation cannot be discussed at pre- 
sent at any fundamental level, since no model violating CPT invariance can 
be constructed that satisfies the requirements of Lorentz invariance and 
minimum analyticity. However, one could discuss the possibility that the 
usual decay formalism still holds phenomenologically, with the two eigen- 
modes \K s y and \K L } in eq. (2) given instead by 

(5) J \Ka>=p'\K> + q'\K>, 

In such a case the two states with zero strangeness are, instead of (4a) and (4b) : 
(6a) \SS>-?£ \LL} + \ g- J) QLS} + \SL» = 

= i(fP^W (|ja> + |K>)> 

(6b) \LSy-\SL} = (pq'+ qp >)(\KKy-\KKy) . 

Equation (6a) exhibits the possibility of observing an interference between 
\SSy and \LS} term , which is present only when 

(7) q -- P - = 2(S^Q, 

q p 

implying violation of CPT invariance. To search for such an interference 
it is probably best to look for the decay (tcCv) at t x togetehr with the decay 
(7T+7T - ) at t 2 from a single pp annihilation. The probability of such events 
depends on t x and t x as follows: 

(8) exp [-V 2 ]|exp [— \(l 8 — 2i Am) t x ] +£exp [— lX L t x ]\ 2 . 



The K°-K° system in p-p annihilation at rest 175 

The interference term would yield a measurement of fi. To appreciate the 
difficulties of the experiment let us assume that |^|~2xl0~ 3 . The most 
bothersome background is then due to the decay of (6b) into the same state 
with time dependence: 

(9) ~ exp [— A s t 2 ] exp [— 1 L t x ] . 

While the time dependence on t x can serve to separate (9) from (8), the small 
magnitude of ft makes the separation possible only with enormous statistics. 
It is doubtful that the experiment could be done with fewer than 10 n K 
pairs or 10 13 stopped p. On the other hand, if CPT invariance should be 
violated, the essential parameter ft seems measurable only in an experiment 
of this type. 



REFERENCES 



[1] Previous discussions of this topic can be found in M. Goldhaber, T. D. Lee and 
C. N. Yang: Phys. Rev., 112, 1796 (1958); and ref. [7]. 

[2] For a previous analysis see V. L. Lyuboshitz, E. O. Oknov and M. I. Podgoret- 
skii: Sov. J. Nucl. Phys., 6, 907 (1968). 

[3] From a practical point of view it may be desirable, in order to enrich the sample with 
K°-K° events, to surround the hydrogen target with anticoincidence counters. To ob- 
serve K s decays high precision is needed, such as is already obtainable with streamer 
chambers. To observe K L decays, very large volume detectors would be needed. 

[4] T. D. Lee, R. Oehme and C. N. Yang: Phys. Rev., 106, 340 (1957). 

[5] See review in Proc. Vienna Conf. on High-Energy Phys. (1968). 

[6] See, e.g., C. N. Yang: Brookhaven Lecture (October 1965) in Brookhaven National 
Laboratory Lectures, vol. 3 (New York, 1965). 

[7] B. d'Espagnat: Nuovo Cimento, 20, 1217 (1961); M. Schwartz: Phys. Rev. Lett., 
6, 556 (1961); G A. Snow: Phys. Lett., 1, 213 (1962); R. Armenteros et ah: Proc. of 
CERN Conf. on High-Energy Physics (1962). 

[8] Unlike a number of experiments under progress on the phase angle of r) + _ which suffer 
somewhat from a lack of knowledge of the precise K°-K° composition, this approach 
is free of such problems. 



Symmetry Principles in Physics. 
T. D. Lee 

Columbia University - New York, N. Y. 



1. - Introduction. 

Symmetry considerations have played an important role since the beginning 
of physics. A great deal of our understanding of nature can be formulated 
through symmetry principles. However, especially over the past decade, we 
have learned that many of these symmetry considerations turn out to be not 
strictly correct. Why do natural laws have a connection with symmetries; 
but often with a slight amount of asymmetry rather than with perfect sym- 
metry? For example, physical laws are almost symmetrical with respect to 
left and right. However, in the weak interactions where 

weak forces 6 

strong forces 

the right-left symmetry is broken. Similar symmetry breaking effects have 
been observed for many other trasformations ; these are summarized in 
Table I. 

Table I. - Symmetry violations in strong interaction processes (e.g., np-^np). 

Violation amplitude 
Symmetry (relative magnitude) 

P (space inversion) 10~ 6 

C (particle anti-particle conjugation) 10~ 3 - 10 -6 

T (time reversal) 10" 3 - lO" 15 

CP 10- 3 - 10" 15 

SU 2 (iso-spin) 10" 2 

su, 10- 1 



Symmetry principles in physics 111 

This situation may appear to some people as aesthetically disturbing. 
Why should nature be slightly asymmetrical? 

Before starting the technical discussion of these symmetries and asym- 
metries in physics, let us recall that the word « symmetry » has two different 
meanings even in our daily language. According to the 1949 Webster's 
Dictionary (*), these two definitions are: 

syrn'me-try (sfm'£.trf), n. [F. or L.; F. symmStrie (now symStrie), 
fr. L., fr. Gr. symmetria, fr. syn- + metron a measure.] 1. Now 
Rare. Due or balanced proportions; beauty of form arising from such 
harmony. 2. Correspondence in size, shape, and relative position, of 
parts that are on opposite sides of a dividing line or median plane. 

It is of interest to note the italic Now Rare for the first definition. Can it be 
anticipating the broken symmetry principles that physicists have now dis- 
covered about nature? Indeed, as we shall see, perhaps beauty should be 
associated with a slight asymmetry, rather than with total symmetry. 

The concept of beauty is, of course, quite subjective. Which is a more 
beautiful object, one with total symmetry, or one with a slight asymmetry? 
The answer is clearly open to debate. However, we may take a look at some 
of the well-known art pieces. For example, both the Greek statue and the 
mosaic from S. Apollinare in Classe near Ravenna, shown in Figs. 1 and 2, 
emphasize bilateral symmetry. The painting of Poplars by Monet in Fig. 3 
suggests a discrete space-translational symmetry. While the beauty of sym- 
metry is clearly demonstrated in each case, this beauty is greatly enhanced 
by the presence of slight asymmetries. 

In physics, our concern is with the laws of nature. The concept « beauty » 
comes in only because of our belief that nature is beautiful. The concepts 
« symmetry » and « asymmetry » are applied to the various transformations 
which are connected with both space-time co-ordinates and the interactions 
between elementary particles. Both symmetry and asymmetry should be 
formulated in terms of precise mathematical language, so that their impli- 
cations can lead to predictions which can be tested experimentally. 

There are four main groups of symmetries, or broken symmetries, that 
are found to be of importance in physics. 

1) Permutation symmetry: Bose-Einstein and Fermi-Dirac statistics, 

2) continuous space-time transformations: translations, rotations, accel- 
erations, etc., 



(*) Webster's New Collegiate Dictionary, 1949 edition (G. and C. Merriam Co., 
Springfield). 



12 



178 



F. D. Lee 




Fig. I. - Praying Boy, Greek sculpture. Reprinted lYcm H, WtYL: Symmetry 
{Princeton University Press, Princeton, New Jersey, 1952). 



Symmetry principles in physics 179 




Fig. 2, - Mosaic from S. Apollinarc in Clas.sc. 




Fig. 3. - Poplars by Claude Monet (on display at the Metropolitan Museum of Art). 



180 T. D. Lee 

3) discrete transformations : space inversion P, time reversal T, particle- 
antiparticle conjugation C, G-parity, etc., 

4) unitary transformations: 

t/ r symmetries : conservation laws of charge Q, baryon number N, 
and lepton numbers L e and L^, SU 2 (isospin) symmetry, and SU 3 sym- 
metry. 

Among these, the symmetries connected with the first two groups of transfor- 
mations are, at present, believed to be exact. In the third group only the 
product CPT is perhaps exact, but each individual discrete symmetry oper- 
ation is not. In the fourth group only the t/ r symmetries are exact. 



2. - Symmetries, nonmeasurables and conservation laws. 

The root af all symmetry principles in physics lies in the assumption that 
it is impossible to measure certain basic quantities; these will be called non- 
measurables in the following discussion. For example, we may consider the 
interaction energy V between two particles at positions r x and r 2 . The physical 
assumption that it is not possible to measure an absolute position leads to 
the mathematical conclusion that the interaction energy V should be un- 
changed under a space translation 



and 



Therefore, the interaction energy V is a function only of the relative distance 

0"l— r 2)» *'•*•> 

(1) V=V(r 1 -rJ. 

From this, we deduce that the total momentum of this system of two par- 
ticles must be conserved, since its rate of change is equal to 

_(V 1 + V a )K 

which, due to (1), is zero. 

This simple example illustrates the close connection between three aspects 
of a symmetry principle: the assumption of a nonmeasurable, the implied 
invariance under the connected mathematical transformation and the physical 
consequence of a conservation law. In an entirely similar way, we assume 



Symmetry principles in physics 181 

absolute time to be a nonmeasurable ; the physical laws must then be invariant 
under a time translation 

t->t + r 



which results in the conservation law of energy. By assuming absolute direc- 
tion to be a nonmeasurable, we derive rotation invariance and obtain the 
conservation law of angular momentum. 

Similar reasoning extends to all other symmetry considerations. The 
special theory of relativity assumes that absolute velocity is a nonmeasurable, 
and the general theory of relativity assumes that the difference between an 
acceleration and a gravitational field is not a measurable quantity. The 
permutation symmetry rests on the impossibility of observing any difference 
between identical particles. 

In order to derive the conservation law of electric charge, we assume that 
it is not possible to measure the relative phase between states of different 
charges; therefore, one must have invariance under an arbitrary multiplica- 
tion of phase factor e id between states of different charges. Since e id is a 
simple lxl unitary matrix, this invariance is called « L^-symmetry ». It im- 
plies that the transition matrix elements between states of different charges 
must be zero, for otherwise there could be interference between two states 
of different charges, and their relative phase would be measurable. This 
E/j-symmetry then leads to the well-known conservation law of electric charge. 
Similarly, the impossibility of measuring relative phases between states of 
different baryon numbers implies the conservation of baryon number; the 
assumption that relative phases between states of different lepton numbers 
are nonmeasurables results in the conservation of lepton number. 



3. - Symmetry violations. 

Violations of symmetries arise when what were thought to be non- 
measurables turn out to be actually measurable. Let us take as a first example 
the question of right-left symmetry. 

The concept that nature (i.e., physical law) is symmetrical with respect to 
right and left dates back to the early history of physics. Of course, in our 
daily life left and right are quite distinct from each other. Our hearts, for 
example, are usually on our left sides. The word « right » means also correct, 
while the word <•< sinister » in its Latin root means left. In English, one says 
right-left, but in Chinese, j£ (left) always precedes fa (right). However, 



182 T. D. Lee 

such asymmetry in daily life is attributed to either the accidental asymmetry 
of our environment or the initial condition in organic life. 

The principle of the symmetry between right and left has been found to 
be true in classical physics, in atomic physics, and in many areas of nuclear 
physics. Yet, in 1957 it was discovered that the laws of nature are in fact not 
symmetrical with respect to right and left. The apparent symmetry previously 
found in macroscopic physics, atomic physics, and nuclear physics is only 
an approximate one. For example, the neutrino emitted in a 7r+ decay has 
its spin always antiparallel to its momentum. Although the initial iz + , being 
of zero spin, is in a totally spherically symmetrical state, nevertheless we may 
use its final decay product to give an absolute definition of left versus right, 

The same applies to other violations of symmetries. Previously, in electro- 
magnetic theory, the sign of electric charge could only be defined in terms of 
that of a test charge. Now, because of C and CP asymmetries, one may give 
an absolute definition of the sign of electric charge. For example, in the 
decay of the long-lived neutral K meson, K^,, the decay rates [1, 2] to e+ and [i+ 
are different from those to e~ and yr: 

rate (K° r -> e + 7r-v ) 

, ; * -^= 1.003 15 ±0.0003 

rate (K" -> e~7r; + v e ) 

and 

rate(K°^ [ i.-7i+v (i ) 

Such slight differences in decay rates enable one to give an absolute definition 
of the sign of electric charge, without the use of a test charge. 

As discussed in the previous Section, the validity of all symmetry principles 
rests on the theoretical hypotheses of nonmeasurables. Some of these hy- 
potheses may indeed be correct in a fundamental sense, some may simply be 
due to the limitations in our present abilities to measure things. As we improve 
our experimental techniques, our domain of observations naturally becomes 
enlarged. It should not be too surprising that we may even succeed in ob- 
serving some of those supposedly nonmeasurables, and therein lies the root 
of symmetry breaking. 

In this sense, we should be prepared for the eventual possibility that we 
might be able to measure absolute space-time positions, absolute direction, 
and absolute velocity, and even relative phases between states of different 
charges, different baryon numbers, and different lepton numbers. Even if 
these were possible, it should be expected that such discoveries could lead 
only to small symmetry breakings in all of the presently known physical 



Symmetry principles in physics 183 

phenomena, because, otherwise, these supposedly nonmeasurables would have 
been measured long ago. 

Just as in most of our artistic creations, the harmony and beauty of sym- 
metry is always enhanced by the presence of a small degree of asymmetry. 
From an aesthetic point of view, it is rather satisfying to find nature also has 
a similar preference in small symmetry violations. 



REFERENCES 



[1] J. Steinberger: Proc. CERN Topical Conf. Weak Interactions (1969), p. 291. 
[2] D. Dorfan et ah: Phys. Rev. Lett., 19, 987 (1967). 



The Design and Use of Large Electron Synchrotrons. 

A. W. Merrison 

Daresbury Nuclear Physics Laboratory - Daresbury, U. K. (*) 



1. - Introduction. 

The large proton synchrotrons built in the last 20 years have been hugely 
successful in their contributions to knowledge in elementary particle physics. 
This has been due primarily to the fact that they serve as intense sources of 
secondary particles in the giga electronvolt region. 

The contribution of the large electron accelerators has not, until recently, 
been anything like as striking as that of the proton accelerators, but now 
that many electron accelerators are working in the giga electronvolt range 
this is a situation which has changed, and is changing, rapidly. It is very 
clear that the large electron accelerators can be used to investigate phenomena 
in a way quite complementary to the proton accelerators. 

There is a fundamental peak energy limitation for large cyclic accelerators, 
first pointed out by Iwanenko and Pomeranchuk which, while quite unim- 
portant for proton accelerators, is critical to the building of electron accel- 
erators. This is the classical electromagnetic radiation emitted by a charge 
undergoing acceleration — in the case of an accelerator, the centripetal accel- 
eration caused by the charged particle following a curved orbit. It can easily 
be shown that the energy loss Ai? (in MeV) of a charged particle of energy 
E (in MeV) in one complete orbit of a circle of radius R (in meters) is given by 



AE 



An e 2 1 E \ 4 



where e is its charge and m its rest mass. For electrons this reduces to 

A£=8.84xl0- 14 ^-. 
R 



(*) Present address: University of Bristol, Bristol, U.K. 



The design and use of large electron synchrotrons 185 

As a practical example, the 5 GeV electron synchrotron NINA has a 
bending radius of 21 m which leads to an energy loss at the maximum energy 
of 2.6 Me V/re volution. But since this radiation loss occurs inversely as the 
fourth power of the rest mass it is completely negligible for all practicable 
proton accelerators. 

It can be shown also that if the magnetic guide field varies sinusoidally 
with time, which is usually the case with electron synchrotrons, then the 
mean energy loss (in watts) W through synchrotron radiation, if the mean 
accelerated current of electrons is / (in amperes), is given by 

FF= 5.78x10-*^, 
jR 2 

where / is the repetition frequency in cycles/sec. This shows clearly that 
where the synchrotron radiation is a serious consideration it can be reduced 
by increasing the repetition frequency of the synchrotron and its radius. 

Apart from the difficulty of providing sufficient radio-frequency power to 
overcome this loss and then to accelerate the electrons, a more fundamental 
limitation is created by this so-called synchrotron radiation. Since the elec- 
tron loses energy not continuously but by the discrete emission of photons, 
synchrotron oscillations are created which can lead to serious difficulties with 
the available magnet aperture for practical accelerators. However, for the 
accelerators discussed in this paper this is not likely to be a serious limita- 
tion, and schemes to reduce it have been put forward by, for example, 
Robinson. 

Of course, none of these considerations applies to linear accelerators and 
this is why the largest electron accelerator built so far, the 20 GeV accelerator 
at Stanford, is a linear machine. Such accelerators, since they are not cyclic, 
suffer the serious disadvantage that their duty cycle is very short; in the case 
of SLAC, for example, it is 0.05 % . This limits the class of experiment which 
can be carried out with this kind of accelerator, though it is entirely possible 
that this difficulty will be overcome by the current development of super- 
conducting radio-frequency cavities. This limitation in turn has concentrated 
a great deal of study on the possibility of building large electron synchro- 
trons, which have the virtue of a good duty cycle. 



2. - What is the maximum practicable energy of an electron synchrotron» 

This question has been discussed in some detail by Crowley-Milling [1] 
and some of his simple arguments will be presented. 

We can get an absolute limit to the maximum energy in a synchrotron 



186 A. W. Merrison 

of given configuration by setting the radiated energy loss per revolution equal 
to the energy gain per revolution. 

In a conventional synchrotron the perimeter is made up of guide fields, 
where the radius of curvature of the electrons is R, and straight sections 
which contain other ancillary equipment — such as, for example, the accel- 
erating cavities — leading to a « mean radius » r for the synchrotron. The 
maximum energy gain per revolution is given by 



(-*)■ 



(1) A2s G = 2nrer I 1 J k cos y , 

where s is the maximum electric accelerating field, r is the transit time factor, 
k is the fraction of total straight length available for acceleration, and (p is 
the synchronous phase angle. By setting this equal to the synchrotron radia- 
tion loss at peak energy and using practicable values for the parameters in (1) 
of t = 1, Rjr = 0.6, k = 0.8, cos 99 = 0.7 and e = 1 Mv/m we arrive at the 
peak energy E 

E = 1.78 xlOM. 

Again as a practical example for r = 1.2 km (which is the radius of the 
proposed European 300 GeV proton synchrotron) we have E = 62 GeV. 

Another limitation discussed by Crowley-Milling is that given by the fact 
that the radio-frequency losses in the normally conducting structures usually 
used as radio-frequency accelerators increase rapidly as the energy increases. 
If we say that the electric field required to maintain the particles in acceler- 
ation varies as E* then clearly the radio-frequency power required to set-up 
these fields will vary as E 8 . Again by making reasonable assumptions about 
the overall accelerator structure the dependence of the maximum energy on 
available radio-frequency power can be deduced. 

If one assumes that it would not be unreasonable to use 10 MW of radio- 
frequency power in such an accelerator the peak energy from an electron 
synchrotron with parameters similar to those used above (including a mean 
radius of 1.2 km) will be about 45 GeV. Since most of this power would 
still be absorbed in the accelerating structure, beam loading would not be 
a serious problem and it looks feasible to accelerate mean currents of up 
to 10 [aA. 

3. - The Daresbury « Booster ». 

For some time now a certain amount of study has been given to the 
possibility of using the 5 GeV electron synchrotron NINA as the injector 
to a very much larger synchrotron in the (15^-20) GeV range [2]. This was 



The design and use of large electron synchrotrons 



187 



prompted by a number of considerations: the clear need in the future for 
a long duty-cycle accelerator in this energy range ; the high cost of the injector 
for such a machine; and the fact that the topology of the Daresbury site is 
well suited to the construction of the main accelerating ring of the large radius 
required if the radio-frequency power is to be kept within acceptable limits. 
By the time the electrons in NINA have been accelerated to 2 to 3 GeV, 
the emittance and energy spread of the beam are very small and this means 




Scale 



Fig. 1. - Layout of Daresbury site for the NINA Booster. 

tunneling. 



represent 



188 A. W. Merrison 

that the magnet aperture of the main accelerator (the « Booster ») can be 
made correspondingly small. Again even with a large radius, the high injec- 
tion energy means that the magnetic field in the main ring at injection can be 
very high. Since in this case much larger remanent and eddy current fields 
can be tolerated at injection this in turn means that one can contemplate 
quite simple forms of construction for the vacuum chamber. 

One possible layout which has been studied for the NINA Booster is that 
shown in Fig. 1. This particular layout has the advantage of making the 
maximum possible use of the experimental facilities, including the existing 
NINA experimental hall, on the Daresbury site. It can be seen too from this 
layout how much of the main ring would lie in a tunnel through the hillside 
on which NINA has been built. 

The general form of the booster would be that of a superperiod machine 
with 4 « quadrants » each on a mean radius of about 1 50 m with long straight 
sections, each of 100 m, separating them. Two of these straight sections 
would be sufficient to house the radio-frequency accelerating sections required 
for acceleration to 20 GeV. A list of the leading parameters is given in 
Table I. 

Table I. 
Parameter Units 

Peak energy 15/20 GeV 

Mean current 1/3 uA 

Mean radius 146.9 m 

Bending radius 120.0 m 

Magnetic field on equilibrium orbit (20 GeV) 0.56 T 

Length of full magnet 7.25 m 

Total number of full magnets 104 

Betatron oscillations per turn 17.75 

Length of straight section 100.0 m 

Magnet excitation frequency 53 Hz 

Radiation loss/turn 20 GeV 118 MeV 

15 GeV 38 MeV 

Radio frequency power (mean) at 15 GeV, 1 [xA 95 kW 



It seems quite clear that there are no real problems in building a quite 
conventional synchrotron to accelerate the NINA beam to 15 GeV with 
perhaps 1 \lA (mean) of accelerated electrons. Before embarking on a second 
stage which would take 3 jxA, say, to 20 GeV it is certain that the possibility 



The design and use of large electron synchrotrons 1 89 

of superconducting radio-frequency structures must be investigated thoroughly. 
The preliminary cost estimates of the first stage are £ 3.82 million, with an 
additional £ 0.7 million required for the second stage. 



4. - Experimental use of a 70 GeV electron synchrotron. 

It is important to bear in mind in thinking about possible experiments 
for a high energy electron synchrotron that for the most part they will be 
experiments which it will be impossible to do with a proton synchrotron of 
comparable or higher energy. There are certain experiments which would be 
done on either kind of accelerator and for which the electron machine will 
offer certain advantages, but these form rather a special class. 

As an illustration of this one can look at the secondary particle yields 
worked out by P. G. Murphy [3] for the NINA Booster at 20 GeV with 
3 [iA of accelerated electrons and compare them with the CERN PS. In all 
cases the Booster yields are a factor of between 2 and 5 down on those from the 
CERN PS. But there are situations where it would be more advantageous to 
use the Booster secondary particle beams. Long-lived neutral kaons, for 
example, are much more free (by a factor of 10 or more) from neutron con- 
tamination, and in addition because of the high radio-frequency used in the 
booster (816 or 1224 MHz) timing information on the kaons can be obtained. 
It must be emphasised however that the principal experiments to be done 
with high energy electron synchrotrons will certainly be those using electrons 
or photons as the primary particle. 

A number of specific experiments which can be done with (15-f-20) GeV 
electrons are investigated in some detail in the report referred to by Murphy 
and Clegg [3]. They have confined their attention to experiments where 
a good duty cycle is essential or, at the least, very desirable. Since the photon 
has a unique combination of quantum numbers (spin, helicity, C) no rest 
mass and a relatively weak interaction there are many experimental situations 
where the photon is an ideal tool to use as the primary particle. This is par- 
ticularly the case when one studies the vector mesons which, leaving aside 
the mass, are rather « photon-like » in their properties. The success of the 
vector dominance model in the strong interaction does not need underlining. 
In the same sense, in inelastic electron scattering experiments, the electron 
can be looked on as a source of « massive » virtual photons. But it is quite 
clear that all such investigations will call for complicated coincidence exper- 
iments and will depend on a long duty cycle accelerator for exploration in 
depth. The field however is extraordinarily rich and can be approached only 
with an accelerator of the kind discussed in this paper. 



190 A. W. Merrison 



REFERENCES 



[1] M. C. Crowley-Milling : Daresbury Technical Memorandum EL/TM/43 (1966). 
[2] M. C. Crowley-Milling and A. W. Merrison: Daresbury Technical Memorandum 

EL/TM/49 (1967); and Daresbury Report DNPL/R2 Preliminary design study for a 

15-20 GeV electron synchrotron: NINA Booster. 
[3] P. G. Murphy and A. B. Clegg (eds.): Daresbury Report DNPL/R3 Experimental 

utilization of a NINA Booster (1969). 



Breaking of the SU 3 x SU Z Symmetry in Hadronic Physics. 

L. Michel 

Inst it ut des Hautes Etudes Scientifiques - Bures-sur-Yvette 

L. A. Radicati 

Scuola Normale Superiore - Pisa 



1. - Introduction. 

In this paper we analyze the properties of the three fundamental inter- 
actions (strong, electromagnetic, and weak) from the point of view of the 
SU 3 xSU 3 group. For this analysis we will use an extension of the geometrical 
approach which we have introduced before [1, 2] for SU 3 . In that case the 
three charges conserved by each interaction namely the hypercharge Y, the 
hadronic electric charge Q H , and the weak hypercharge Z, are generators 
of the unitary linear representation of SU 3 on the Hilbert space of hadronic 
states. That is, in the representation a ~> Q{a) e if(^) of the SU 3 Lie algebra 
on 3t, Q H , Y, Z are the images of three vectors — q, y, z of R 8 , the octet space, 
i.e., the eight-dimensional real vector space of the Lie algebra of SU 3 . As 
we have shown in ref. [1] the isotropy groups of these vectors are maximal 
subgroups of SU 3 and the vectors themselves are solutions of a nonlinear 
equation. 

It is however clear that for a full understanding of the properties of the 
interactions and of their relations we need to consider the group SU 3 x SU 3 . 
Indeed the different behavior under space reflections of the three inter- 
actions, cannot be described in terms of the diagonal SU 3 subgroup alone. 

We will see that some of the interesting geometrical properties of the 
vectors y, q, z can be carried over to SU 3 xSU 3 . We will show that the 
directions along which the symmetry group is broken are, also in this case, 
solutions of nonlinear equations of the type postulated by the bootstrap 
approach to symmetry breaking. 

Two subgroups of SU 3 x SU 3 are of special significance for hadron physics : 
SU 3 and SU 2 xSU 2 . Both represent approximate invariances of the strong 



192 L. Michel and L. A. Radicati 

interactions which are valid when one neglects either the difference between 
the K- and Tc-meson mass (for SU 3 ) or the pion mass (for SU 2 xSU 2 ). 

Recently Gell-Mann, Oakes and Renner [3] have suggested that the strong 
Hamiltonian which breaks the SU 3 x SU 3 symmetry transforms approximately 
like an element of the (3, 3) © (3, 3) representation which is left invariant 
by SU 2 xSU 2 . We will show that in the space of the (3, 3) © (3, 3) repre- 
sentation we can define two directions which are solutions of nonlinear 
equations and whose isotropy groups are precisely SU 3 and SU 2 xSU 2 . 

In Sect. 2 after a brief resume of the relevant results of refs. [1] and [2] 
we will discuss the unique symmetrical algebra, on the space of the (1, 8)© 
© (8, 1) and of the (3, 3) © (3, 3) representations, which have SU 3 x SU 3 as a 
group of automorphism. The existence on these spaces of symmetrical 
algebras insures the possibility of having nonlinear equations whose so- 
lutions define the directions along which SU 3 xSU 3 is broken. 



2. - Mathematical preliminares. 

2'1. Geometry of the octet. - We begin by briefly reviewing a co-ordinate- 
free formulation [1] of the SU 3 invariant algebras on the octet space R 8 . 

We can realize R 8 as the real vector space of all 3x3 Hermitian trace- 
less matrices a, b, c, .... Any element u of the group SU 3 is the form u = 
= exp [— icpajl], aeR 8 . The action of SU 3 on R 8 (which is the space of its 
adjoint representation) is 

(1) a ^> uau* = uaur 1 . 

We can define on R 8 an SC/g-invariant scalar product and two algebras 
which have SU 3 as automorphism group: 

Scalar product: 

(2) (a, b) = \trab 
SU 3 Lie algebra: 

(3) af\b=- l -[a,b}. 
Symmetrical algebra: 

(4) a yb = \{ab + ba) - f (a, b) = \{a, b] - f (a, b) . 

If a and a \Ja are linearly independent they generate a two-plane ^ a {i-e. a two- 
dimensional subspace of R 8 ) which is a Cartan subalgebra {i.e. a maximal 
Abelian subalgebra) of the SU 3 Lie algebra. Thus ^ a which is isomorphic 



Breaking of the SU 3 xSU 3 symmetry in hadronic physics 193 

to U x x U x is the Lie algebra of the isotropy group (or little group) of a. 
If on the contrary 

(5) qWq + fj(q)q = 0, 

the isotropy group is a U 2 group which we denote by U 2 (q) . Any vector whose 
isotropy group is a U 2 will be called a « q- vector ». From now on we will 
consider only normalized « positive » q- vectors, i.e. such that: (q, q)=l, 
fj(q)> 0. This implies, rj(q) = 1/V3- 

The Cartan subalgebras of the SU 3 Lie algebra are all conjugate (i.e. 
transformed into each other) by the SU 3 group. One of them is of course 
made with the diagonal matrices ueSU 3 . It can be proved that any <$ 
contains three positive normalized #- vectors at 120° from each other. Conv- 
ersely if jcjgR 8 commute, ax + /?y and a'x + fi'y commute, and generate 
a ^ (denoted ^x,y)' For positive normalized ^-vectors we thus have 

(6) (qi,qd= — io#A# = and qt¥-qj. 

Given a ^-vector y, the vectors t y of U 2 (y) which are orthogonal to y form 
the SU 2 (y) subalgebra of U 2 (y). They satisfy the following relations: 

(7) y/\t y = y/\t' y = Q\ CM) = CM') = 0; V%\fty = (h,t y )y. 

The normalized t a of the three ^-vectors of a Cartan subalgebra <€ form 
the hexagon of the « roots » different from zero. 

2'2. The SU 3 X SU 3 algebra. - To extend this formalism to SU 3 X SU 3 we 
consider the space R 16 = R 8 © R 8 . We call a + and a_ the elements of the 
first and the second R 8 , respectively (the index ± corresponds in physics 
to the chirality) and denote by a = a + © a_ an element of R 16 . The Lie 
algebra of SU Z x SU 3 is then defined by 

(8) a A I = (a+ © a_) A (b + © fe_) = (a + A 6 + ) © («_ A 6_) , 

where A in the right-hand side has been defined on R 8 in eq. (3). The scalar 
product invariant under SU 3 x SU 3 is the Cartan-Killing form which we write 

(9) (a + ®a_,b + ® b_) = K«+ , K) + Kfl- , *-) • 

It is also convenient to use another decomposition of R 16 into a direct 
sum R 8 © R 8 . In this decomposition, which is symbolically illustrated in 
Fig. 1, we denote the element a = a + ®a_ by (a\a') with 

(10) a + =a-\-a' and a_= a — a'. 

13 



194 L. Michel and L. A. Radicati 

sut' 



antidiagonal 




Fig. 1. - Decomposition of R 16 into a direct sum of R 8 ff)R 8 . 



In this notation the Lie algebra law (6) becomes 

(11) aAb = (a\a')A(b\b') = (aAb + a'Ab'\aAb' + a'Ab) 

and the scalar product 

(12) {a,l) = {a,b) + {a\b'). 

In a similar way we can extend to R 16 the symmetrical algebra on R 8 : 

(13) aVb = (a\a')\/(b\b') = (a\jb^ r a'yb'\a\/b'+a l \jb) . 

One verifies that the equation 

(14) a\/a = Xa, 
has only two types of solutions: 

05) a=(q\0) 

and 

(16) a=(q\±q), 

where q is a ^-vector. 

The subalgebra of SU 3 x SU 3 which leaves invariant (i.e. commutes with) 
a ^-vector (j|0) of the diagonal SU 3 subalgebra is the set of all («|«') such 
that yAa = 0, yAa' = 0; it will be denoted (U 2 \U 2 ) y . With the notation of 
(8) it is the direct sum U ( 2 +) (y) © U 2 -\y) . 



Breaking of the SU 3 xSU 3 symmetry in hadronic physics 195 

2'3. The (3, 3) (3, 3) representation. - A special role in the physical 
applications is played by the (3, 3) representation of the group or of the 
Lie algebra SU 3 x SU 3 . We can realize the 9-dimensional space of this repre- 
sentation as the complex vector space of the 3x3 matrices m. Under the 
transformation u + xu_= exp [— i<pa + /2] x exp [— iya_/2], m goes over into : 

(17) /w~> u + mu* . 
The representation of the Lie algebra is thus 

( 1 8) D(a) m = D(a + a_)m = — l(a + m — ma_) , 
or 

(19) D(a)m = D(a\a')m = —~([a, m] + {a', m}) . 

(Equation (18) is obtained from (17) by differentation with respect to <p, 
at cp = 0.) 

The representation is unitary for the group, i.e. it leaves invariant the 
Hermitian scalar product 

(20) <!»! , m 2 > = | tr {ml m 2 ) . 

The 9-dimensional complex vector space C 3 -^ can be considered as an 
18-dimensional real vector space R 18 . The 1 8-dimensional representation of 
the group SU S x SU 3 on this space is the direct sum of the (3, 3) and its complex 
conjugate (3, 3). It is real and unitary, hence orthogonal. It leaves invariant 
a Euclidean (i.e. real orthogonal) scalar product which is the real part of 
(20), while the imaginary part becomes an antisymmetrical real (i.e. sympletic) 
scalar product. Explicitely we have: 

(21) 0»i,»2) = Re<m 1 ,w 2 > = Itrt/wfma + /n|m 1 ) , 

(22) )m 1 , m 2 ( = Im (m 1 , w 2 > = — tr (m\m 2 — mtm^ . 

Any 3x3 complex matrix can be written in the form 

(23) m = V|ul + m + iVf 1//'+ im' = (ji\m\\fj,'\m') , 

where ju and fx' are real members and m and m' are vectors of the octet space. 
In this notation (21), (22), and (23) read 

(24) (m 1 , m 2 ) = /x lf x 2 + ju[ju' 2 + (m lt m 2 ) + (m' x , m 2 ) , 

(25) )m x ,m 2 ( = fx^ 2 — ^ 2 + (m l9 m 2 ) — (m[, m 2 ) , 



196 



L. Michel and L. A. Radicati 



and eq. (19) reads 

(26) D{a\a')(jj\m\\fi'\m') = 

= (Vf(a', m')\aAm + a'ym' + V|//a'|| — V}(a\ m)\a/\m r —a'\/m — V$fjia') . 

Tensor operators which represent physical observable must be Hermitian 
on Jt. It is therefore necessary that they belong to a real representation 
of the invariance group. This is the case of the (3, 3) © (3, 3) representa- 
tion which, we want to emphasize, is irreducible as a real representation. 

The tensor product of (3, 3) © (3, 3) by itself when decomposed into 
real irreducible representations contains the (3, 3) © (3, 3) once. 

Hence from two vectors r, s e (3, 3) © (3, 3) it is possible to form a new 
vector of the same representation which we denote r T s. The symbol T is 
the law of a symmetrical algebra on R 18 which has SU 3 x SU 3 as automorphism 
group. By standard methods we find 

(27) r T s = l l(trr* trs*— tr (r*s*)) — §#•* trs* — \s* tr r* + |{r*, s*} . 
We leave to the reader to check that 



(28) 



D(a\a')r T s = (D(a\a')r) T s + r T (D(a\a')s) , 



which means that SU 3 xSU 3 is a derivation algebra of the ^-product. 
With the notation (23) we can write eq. (27) in the form 



(29) 
where 

and 



(30) 



r r J = (T|f||T'|f'), 
r = (^|r||o'|r') and s = (o\s\\o'\s') 

r = -=(2QO-2 Q 'o'-(r, s) + (r f , s')) , 



t = —^{—os — or + q's' '+o' 'r')-\-r\Js — r'\ls' 
\/6 



t' = -~(qs'+ q's + o'r + or') — rys'— r'\js 



Breaking of the SU 3 xSU 3 symmetry in hadronic physics 197 

We add two more properties of this product 

( 31 ) <x T y, z> =<x,y T z> , 

(32) <*, x T x} = f detx = (x T x, x) + i)x T x, x( . 

The (3, 3) © (3, 3) representation of SU 3 x SU 3 has no invariant for the sub- 
group (U 2 \U 2 )y, i.e. if for all (a\a')e(U 2 xU 2 ) y , D(a\a')m = eq. (26) 
shows that m = 0. However if a' is restricted to be in SU 2 (y), the same equa- 
tion shows the existence of a two-dimensional invariant subspace spanned 
by the vectors 

(33) (v\-V2ny\W\-V2n'y), 

where y is a ^-vector and r\, r\' are real numbers. We will denote the isotropy 
group (or its Lie algebra) of the vectors (33) by (U 2 (y)\SU 2 (y)). This Lie al- 
gebra is the following direct sum 

(34) (U 2 (y)\SU 2 (y)) = SUpty) © SU^iy) © U*(y) , 

where Uf(y) = (Ux(y)\0) is the Lie algebra generated by (y\0) (see Fig. 1). 
The vectors (33) have an interesting property under the T-product. Let 
y((p) be the vector (33) with 

(35) tj =V^coscp , rj' = VJ sin^ . 
These vectors are normalized 

( 36 > 0#),J#)> = 1. 

They belong to the SU 3 xSU 3 orbit oi>(0) and satisfy the quadratic equation 

(37) yQp) T y(<p , ) = o. 

Moreover one shows that all unit vectors of R 18 satisfying such an equation 
are on the SU 3 x SU 3 orbit of y(0). 

Equation (37) is a particular case of the equation 

(38) m T m = Am . 

If I =£ (and m ^ 0) one also shows that the only unit vectors which are 
solutions of (38) are on the two orbits of ±n=± V2/3 1 which are SU$ 
invariants. The unit vectors which have SU$ as isotropy group form a circle 

(39) 11(99) = Vf exp [i(p]l = (cos<p|0||sin<p|0) 
n{cp ±n)= — n((p) 



198 L, Michel and L. A. Radicati 

and they generate the T-subalgebra 

(40) <<P) T <<P') = ^ln(-cp-cp>) . 

3. - Geometrical properties of the three interactions. 

3*1. The SU 3 symmetry. - We begin by recalling the basic properties of 
the interactions under the SU 3 group, i.e., the diagonal subgroup SUf of 

sui +) xsui-\ 

a) The hypercharge Y and the three isospin operators T 1 ,T 2 ,T 3 = 
— Q H + \ Y generate the invariance group U 2 (y) of the strong interactions. 
The extension of this invariance to SU 3 implies considering U 2 (y) as a sub- 
group of SU 3 . This means that y is a q- vector of which Y is the image in the 
representation of the SU Z algebra in the Hilbert space Jf of hadron physics. 

The electric hadronic charge Q H is the corresponding image of — q and 
the relation Q n =T z -\-\Y implies that q is a #- vector. The SU 2 (q) group is 
called the « [/-spin group ». 

b) According to Cabibbo's hypothesis the two charged components of 
the vector current v± coupled to the leptons and the electromagnetic cur- 
rent j e - m ' belong to the same SU$ octet. We denote by c 1= bz'c 2 the directions, 
of y±. Using the additional property that the electric charges of v± are ± 1 
we can deduce 

(41) y/Zc-i V c 1 = V3c 2 V c 2 = z , 

where z is a ^-vector. The operator Z, which is the image of z, is the weak 
hypercharge conserved in weak interactions. 

The vector z commutes with q but not with y. We thus have in R 8 two 
distinct algebras ^ ay and %> qz which have q in common. The noncommuni- 
cativity of y and z reflects the existence of strangeness violating weak inter- 
actions. As one can see from (6) the difference from 0° or 120° of the angle 
between y and z gives a measure of the noncommunicativity of Y and Z 
and is therefore related to the Cabibbo angle 6. Explicitely we have 

(42) (j,z)=l-fsin 2 0. 

It can be proved that two noncommuting g-vectors y and z uniquely define 
another ^-vector which commutes with both of them. This vector is given 
by the relation 

(43) q = ((y, z)- \)-\<fiy\/z + \{y + z)) . 



Breaking of the SU 3 xSU 3 symmetry in hadronic physics 199 

Thus the strong and weak interactions determine uniquely the direction 
of the electromagnetic interactions. 

Cabibbo has also postulated that the axial currents af belong to another 
SU% octet in the same directions c 1= b/c 2 as v±. The two assumptions about 
the vector and the axial vector currents are in good agreement with experiment. 

3'2. SU 3 xSU 3 symmetry. - Since the weak interactions have a definite 
(negative) chirality whereas the electromagnetic and strong interactions have 
a defined parity, their relations can only be fully understood by considering 
the enlarged group SU 3 xSU 3 . It has indeed been suggested [3, 4] that this 
group and its subgroups provide a reasonably approximate frame for the 
study of hadron physics. Cabibbo's hypothesis can be generalized to SU 3 x SU Z 
by assuming that j*- m ; v± , a± belong to the same representation of this group 
namely the adjoint representation (8, 1) 0(1, 8). We can thus write for the 
currents : 

(44) #». = h£q\0) ; v± = ^(c±|0) ; fl ± = ^(0|c±) . 
The weak currents are thus 

(45) /z± = ^(c±|-c±). 

As Q H is the integral over space of the time component of j e - m -, the integrals 

(46) Q(a\a') = (d 3 x h (a\a') , 

are at a given time the generators of SU 3 xSU 3 . We shall now list the co va- 
riance properties of the three interactions under SU 3 x SU 3 : 

a) The isotropy group of the electromagnetic current hj^q\t$\ and 
therefore of the electromagnetic interactions (see Sect. 2*2) is (U 2 \U 2 )q = 
= Ui + \q)®Ui-\q). 

b) The pair of weak currents hjic^ — c±) and therefore the semi- 
leptonic weak interactions have for isotropy group SU^ U[~\z). 

c) The co variance of the CP conserving Hamiltonian M ? NL for non- 
leptonic weak interactions is not yet established. If it involves only charged 
currents as many physicists would prefer [5] then it would have components 
outside the (1, 8)0(8, 1) representation. It is however compatible with the 
present evidence to assume that ^ NL belongs entirely to the representations 
(1, 8)0(8, 1) in the direction (z\—z) [6]. If this were the case the isotropy 
group of M" NL would be SU 3 +) © U^~\z) which is a maximal subgroup of 
SU 3 xSU 3 . Nothing is known for the CP violating part. 



200 L. Michel and L. A. Radicati 

d) We have said that U 2 (y) and SU$ are approximate invariances of 
the strong interactions. Another interesting approximate invariance has been 
recently proposed by Gell-Mann, Oakes, and Renner [3]. According to 
them, in the limit where the pion mass can be neglected the strong Hamilto- 
nian is of the form 

(47) jf s = jr + jr(/ft), 

where J^ is invariant under SU 3 xSU 3 and 3^{ni) transforms like the 
(3, 3) © (3, 3) representations. They also suggested that to a good approxi- 
mation m coincides with the vector y(0) of eq. (37). 

In this model the approximate isotropy group of the strong interactions 
would be (U 2 \SU 2 ) y which is a maximal isotropy group for the nonzero vector 
of the (3, 3) © (3, 3) representation. 

Even though the mass difference m K — m n is larger than m n , SU 3 remains 
an interesting approximation for the strong interactions. The SU 3 xSU 3 
breaking part in eq. (47) is in the SU 3 invariant direction denoted by «(0) 
in eq. (39). It is remarkable that its isotropy group (SU$) is the other max- 
imal isotropy group of the nonzero vectors of the (3, 3) © (3, 3) represen- 
tation. 

3'3. SU 3 xSU 3 and space reflections. - In the limit where they are exact 
the U 2 (y) and SU§ symmetries of the strong interactions commute with the 
Poincare group (without time reflection). 

For the exact SU 3 x SU 3 symmetry the invariance group is no longer a 
direct product of the internal symmetry group by the Poincare group but 
the following semidirect product 

(^ x^x^-))xZ 2) 

where ^ is the connected Poincare group and the nontrivial element r of Z 2 
acts on ^ like the space inversion and interchanges SU^ with SU^ +) . The 
action of r on the (8, 1) © (8, 1) representation is 

(48) (a\a')^(a\—a'). 

This allows to assign a parity to the elements of the (8, 1)0(1,8) repre- 
sentation; the primed vectors have odd parity, the unprimed ones have 
even parity. 

For the (3, 3) © (3, 3) representation, eq. (26) shows that the primed 
and unprimed quantities which appear in (23) have opposite parity. For 
example j(0) and y(7t/2) (see eq. (33)) are eigenvectors of r with opposite 
parity. As we have seen, SU 3 x SU 3 implies the existence for this represen- 



Breaking of the SU 3 xSU 3 symmetry in hadronic physics 201 

tation of the T-algebra and this fixes in the SU 3 limit the assignment of the 
parity. Indeed, as we have shown, the direction along which SU 3 xSU 3 is 
broken in an St/g-invariant way satisfies the nonlinear equation 

(49) n(0) T n(0) = ]/| «(0) = n^j T n (^j . 

Thus under r 

(50) m = (ji\m\\fi'\m') ^(ji\m\\—iA , \—m'') . 

4. - Remarks on symmetry breaking. 

It has been suggested by several authors [7] that the SU 3 or the SU 3 x SU 3 
symmetries are spontaneously broken. Such a symmetry breaking occurs 
when the invariance group K of a stable state of a physical theory is only a 
subgroup of the invariance group G of the theory itself. In this case all 
states of the same orbit GJK of solutions are all stable states. 

We have shown [1, 8] that in a theory based on a variational principle 
spontaneous symmetry breaking can occur and one expects the subgroup K 
to be a maximal isotropy group among those of all possible orbits. As we 
have seen the breaking of SU 3 x SU 3 by the strong interactions has the above 
property both in the SU 3 or in the SU 2 x SU 2 approximation [9]. The same 
is true of Jt NL if its invariance group is SU^ +) @U^~\z) (see Sect. 3'2). This 
may therefore suggest that the SU 3 xSU 3 symmetry of the hadronic world 
is spontaneously broken by the strong and perhaps also the weak inter- 
actions. 

The intersection between the two isotropy groups of the weak nonleptonic 
nteractions and of the strong interactions in the Gell-Mann, Oakes, and 
iRenner model is: 

(51) (SU^ © Ui-\z)) n (SUi + \y) ® SUtXy) tf?G0) = SUf\y) £/<"%) , 

where q is a ^-vector commuting with y and z which, as we have seen, is 
uniquely defined once y and z are fixed. The intersection of the two groups 
in the left-hand side of (51) and SUi is Uf(q) which is thus the only invariance 
group for the interactions between hadrons (when the hadron-lepton inter- 
actions are disregarded) and corresponds to the conservation of the electro- 
magnetic charge. We have thus the following scheme of decreasing inva- 
riance inside the hadronic world 

O (U 9 \SU a )y d 
SU 3 XSU 3 UMDU^q). 

o SUi 1 



202 L. Michel and L. A. Radicati 

Let us remark that the isotropy group of the electromagnetic Hamiltonian 
is (U 2 \U 2 ) q . This group is not maximal in SU 3 xSU 3 for the (1, 8) ©(8, 1) 
representation. 

However the direction (q\0) of the electromagnetic interactions shares 
with the directions of the two other interactions, i.e. (z\—z) and y(0) or 
«(0) the following properties: they are the different types of solutions of 
SU 3 xSU 3 invariant nonlinear equations: 

(52) (a\a')v(a\a , ) = A(a\a f ), 
for (q\0) and (z|-z); 

(53) m T m — lm , 

for the two directions along which SU 3 xSU 3 is broken with approximate 
SU 3 or SU 2 invariance. Bootstrap approaches to symmetry breaking lead to 
this quadratic type of nonlinear equations. 

It is interesting to note that for an SU 3 x SU 3 invariant theory, the space 
inversion operator r can only be denned modulo an inner SU 3 xSU 3 auto- 
morphism. However, as we have discussed in Sect. 3'3, the existence of the 
T-algebra fixes naturally the parity of the vectors of the (3, 3) © (3, 3) rep- 
resentation of SU 3 x SU 3 and the vector w(0) has even parity. Thus the 
requirement that the breaking due to strong interactions satisfies eq. (53) 
fixes the parity of the hadronic states. 

It is also worthwhile to point out that one of the solutions of eq. (52), 
(z|— z) has a pure chirality corresponding to maximal violation of the 
parity fixed by the strong interaction. The other solution (q\0) has a definite 
parity and its direction q is fixed when j(0) and (z|— z) are known. From 
the point of view of SU 3 x SU 3 the three directions according to which the 
symmetry is broken have thus fairly simple properties and correspond to 
all three types of solutions of the nonlinear equations (52) and (53). 

There is nothing however to tell us why the directions y and z should 
make precisely the angle that is experimentally observed. 

We do not want to discuss here the attempts [10, 11] to calculate 6. We 
only remark that in the SU 3 xSU 3 scheme, m and (z|— z) are not in the 
same representation space. Thus an SU 3 x SU 3 invariant depending on these 
two vectors has to be at least quadratic in m. For example if we define the 
vector (see eq. (18)) 

v = D(z\ — z\i 



z)m 



we can form an SU 3 x SU 3 invariant <«, v} which is a function of (y, z). 



Breaking of the SU 3 x SU 3 symmetry in hadronic physics 203 

However the length of the vectors has not been given here a physical mean- 
ing as we did not take into consideration the strength of the coupling. 

On the other hand a projective invariant such as f z = (jv T , v) (v, v}~$ 
depends upon both 6 and the matrix elements of m which are functions of 
the physical masses. We note that in the limit where m = j(0) (invariant 
under (U 2 \SU 2 )y, f z vanishes. 



REFERENCES 



[1] L. Michel and L. A. Radicati: Proc. Coral Gables Conference, 1968. 

[2] L. Michel and L. A. Radicati: The geometry of the octet (unpublished). 

[3] M. Gell-Mann, R. J. Oakes and B. Renner: Phys. Rev., 175, 2195 (1968). 

[4] For a review of the validity of the SU 3 x SU 3 symmetry and for an extensive list of 
references see S. Weinberg: Proc. 14th Intern. Conf. High-Energy Phys. (Vienna, 1968). 

[5] M. Suzuki: Phys. Rev., 144, 1154 (1966); H. Sugawara: Phys. Rev. Lett., 15, 810 
(1965); M. Suzuki: Phys. Rev. Lett., 15, 896 (1965). See also N. Cabibbo: Proc. 
13th Intern. Conf. High-Energy Phys. (Berkeley, 1967). 

[6] L. A. Radicati: in Old and New Problems in Elementary Particles Physics, edited by 
G. Puppi, p. 272 (Academic Press, New York, 1968). 

[7] For a general discussion see R. E. Cutkowsky: Brandeis University Summer Institute 
in Theoretical Physics, New York, 1965. The spontaneous breaking of SU 3 has been 
specifically discussed by T. Nagy: Nuovo Cimento, 43 A, 654 (1966); R. Brout: Nuovo 
Cimento, 46 A, 932 (1967); N. Cabibbo: in Old and New Problems in Elementary Par- 
ticle Physics, edited by G. Puppi, p. 62 (Academic Press, New York, 1968); L. Michel 
and L. A. Radicati (see ref. [1]). 

[8] L. Michel: in Lectures on Theoretical Physics, 10, 263 (1968), edited by W. E. Brit- 
ten and A. O. Barut. 

[9] This property has been shown on an explicit model in the case of a SU 2 x SU 2 inva- 
riant mass term by G. Cicogna, F. Strocchi and R. Vergara Caffarelli: Phys. 
Rev. Lett., 22, 497 (1969). 
[10] R. Gatto, G. Sartori and M. Tonin: Phys. Lett., 28 B, 128 (1968) and Proc. CERN 

Conf Weak Interaction (January 1969), p. 465. 
[11] N. Cabibbo and L. Maiani: Phys. Lett., 28 B, 131 (1968) and Proc. CERN Conf 
Weak Interaction (January 1969), p. 485. 



Structure of Matter Investigations 
by Thermal Neutrons in Rome. 

A. Paoletti and S. Sciuti 

Comitato Nazionale per VEnergia Nuclear e (CNEN) - Roma 
Istituto Fisico della Facoltd di Ingegneria - Roma 



1. - Introduction. 

As it is well known, the slowing down of neutrons and the interacting 
properties of thermal neutrons with matter were discovered in Rome by 
Amaldi, Fermi, Pontecorvo, Rasetti and Segre [1] thirty-five years ago. 

Since then a tremendous amount of work has been done in all the world 
in the field of nuclear, atomic and molecular physics by means of thermal 
neutrons which at the present time are still successfully employed in many 
first class investigations. 

Until 1955 there was very little in nuclear energy researches going on 
in Italy; particularly in the field of the so-called « pile neutron research » 
every thing was to be started « ex-novo ». 

E. Amaldi had always followed with great interest this kind of research 
and was convinced of the necessity of promoting it in Italy also. Among 
the several lines of research with neutrons he suggested neutron diffraction 
for solid state investigations warmly promoting also the use of thermal 
neutrons in nuclear physics. 

This point of view was completely accepted by the Comitato Nazionale 
per le Ricerche Nucleari [2] leading to the constitution of the Laboratorio 
di Fisica Nucleare Applicata in Rome and of similar units in other Italian 
nuclear Centers [3]. 

In this paper we would like to briefly describe some recent research both 
in nuclear and solid state physics carried out at the Casaccia Center of CNEN, 
Rome, using thermal neutrons. 

The thermal neutrons are provided by the 1 MW, RC-1 reactor [4]. Two 
radial beam tubes are employed for solid state physics work, while two 
tangential beam tubes, one completely crossing the biological shield, are 
used for nuclear physics experiments. 



Structure of matter investigations by thermal neutrons in Rome 



205 



2. - Nuclear physics work. 

The (n, y) reactions are intensively used at the Casaccia Center to study 
nuclear levels in different classes of nuclei. Besides these experiments we 
would like to describe one which makes use of a very fruitful technique 
developed in our laboratory [5]. High energy gamma-lines (5-^8) MeV pro- 
duced from (n, y) reactions in a target located very close to the reactor core, 
come from a beam tube completely opaque to neutrons and impinge on a 
scatterer which is viewed by Ge(Li) and Nal(Tl) large detectors. When a 
gamma-line resonantly excites a high energy level of the target (scatterer) 
nucleus, a gamma-ray cascade takes place originating an inelastic scattering 
process. In this way the physical characteristics of the high energy level 
starting the cascade, together with the lower ones populated by the cascade 
can be efficiently studied analysing the high resolution single spectra, the 
angular distributions, the coincidence spectra, etc. 

In Fig. 1 a sketch of the experimental set-up [6] is shown. In general 
this technique allows one to study several nuclear parameters such as resonant 
cross-sections, ground state and total radiation widths [7], and finally to 
determine the level scheme of nuclei restricted to the transitions starting 




j fn.fiSOUBer " 



Fig. 1. - Nuclear spectroscopy work at the RC-1 Reactor: The experimental 
set-up concerning resonant scattering. 



from the resonant level. This kind of spectroscopy gives information similar 
to those obtainable with (n, y) reactions with the advantage that level spins 
can be determined by measuring the directional correlations of the elastic 
and inelastic gamma-rays; further (y, y') reactions are the most suited for 



206 



A. Paoletti and S. Sciuti 



the study of stable isotopes which may not be reached by (n, y) reactions. 
The choice of the target is considerably restricted by the requirement that a 
random overlap should exist between the incident gamma-line and a level 
in the particular nuclide to be studied. 

Nevertheless about 50 cases of nuclear resonance fluorescence, almost 
near the closed-shell regions, have been since observed [8]. 

We have systematically studied ten nuclei by the nuclear resonant tech- 
nique [7]. Of these the level scheme of 62 Ni, 112 Cd, 118 Sn [6], 205 T1 and 65 Cu [9] 
have been deduced. 

An example of the power of the method in spectroscopy work is given 
in Fig. 2. Monocromatic photons (7646 keV) produced by Fe(n, y) reaction, 
resonantly excite a 205 T1 level. The analysis of all the spectrum shows the 
existence of more than 20 levels with energies between ground state and 
~3.5MeV. In the figure a portion of inelastically scattered gamma-rays, 
detected by a 30 cc Ge(Li) counter, is reported. 



COUNTS 







den 50907 KeV 

/^4rWHM e 17KeV 








d*04899KeV 




■ 


1 






'""■'•■ ' 'Wi%\. 




d«D5333 KeV t "'^^f 


dec54855KeV 


WWk' 


wL 


de.p. 5347KeV s«.p. 4925 KeV 


3.90 





4.000 


4.100 


4200 


4300 4400 


4.500 



E r (KeV) 



COUNTS 



P5425.6 > Hi «925KeV 



iTWHM* 16 KeV 

d e.p.5 773Kf 



8.P5333* 5347KeV 



4mS side 

Ej CKeV) 



Fig. 2. - Nuclear spectroscopy work at the RC-1 Reactor: A portion of the 
(inelastic) spectrum of 205 T1 resonantly excited by Fe(n, y) 7 646 keV y-line. 



In conclusion we would like to point out that nuclear spectroscopy by 
gamma-rays resonant scattering excitation leads to results very similar to 
that achieved by Coulomb excitation and therefore it may be considered a 
very important part of a research program to be performed with reactors. 



Structure of matter investigations by thermal neutrons in Rome 207 



3. - Solid state physics work. 

Neutron diffraction is a powerful tool for studying some microscopic prop- 
erties of solids such as magnetic structures, magnetization densities, lattice 
dynamics, magnetic excitations, phase transitions, etc. 

At the Laboratorio di Fisica Nucleare Applicata research has been per- 
formed in ferromagnetism by use of the polarized neutrons. The static and 
dynamic behaviour of unpaired electrons in metals and alloys has been 
investigated mostly by determining magnetization density maps and disper- 
sion relations of spin waves. So far neutron diffraction has been used at 
Casaccia mostly for 3d electrons [10]; however recently experiments have 
been carried out also on Ad and 5d electrons. 

Transition metals of Ad or 5 d group do not exhibit a magnetically ordered 
state as elements, but some of their alloys with 3d metals show bulk mag- 
netization properties which were interpreted as indicating a contribution of Ad 
and 5d electrons to magnetization. Polarized neutrons provide an unique 
tool for obtaining direct microscopic information and have been successfully 
used both for elastic and inelastic scattering from some Pt alloys, namely 
CoPt 3 , CoPt, and MnPt 3 [11]. It has been possible to ascertain that a fairly 
well-localized magnetic moment of about 0.2 Bohr magnetons is generally 
present on Pt atoms, and magnetization density maps have indicated the 
spatial distribution of 5d electrons and the symmetry of their wave functions. 
These results are appreciably affected by the interaction with neighbour 
atoms as it is proved by their dependence on the order parameter. A typical 
magnetization density around a Pt site in a MnPt 3 alloy with order parameter 
S = 0.22, is given in Fig. 3. One notices that: 

1) The magnetization changes from negative to positive values going 
from the center of the site outward. 

2) The distribution of unpaired electrons is highly aspherical. 

The first result can be consistently interpreted by assuming that the moment 
of Mn atoms occupying the Pt site, because of the incomplete order are 
directed opposite to bulk magnetization. Their negative magnetization due 
to 3d electrons which are fairly concentrated prevails around the nucleus while 
the positive magnetization of 5 d electrons which are more spread out is prev- 
alent away from the nucleus. (One must remember that neutron diffraction 
provides information on the single sites of the unit cell but averaged over 
all the unit cells of the sample.) 

MnPt 3 has been also investigated from the dynamical point of view. The 
acoustical branch of the spin waves dispersion curve has been determined 



208 A. Paoletti and S. Sciuti 

[001] 




-0.02 



Fig. 3. - Neutron diffraction work at the RC-1 Reactor: Three-dimensional 
magnetization map around the Pt site in a MnPt 3 alloy. 



with polarized neutrons by the « diffraction method », which provides such 
information with a simple two crystals experiment. The results have been 
interpreted according to calculations performed at Casaccia [12] on alloys 
of Cu 3 Au type and provided the values for the Mn-Pt and Mn-Mn exchange 
integral in fair agreement with estimations based on the Ising model. 



Structure of matter investigations by thermal neutrons in Rome 209 



REFERENCES 



[1] E. Amaldi, E. Fermi, B. Pontecorvo, F. Rasetti and E. Segre: Ric. Scient.,5 (2), 380 
(1934). For more details on this subject see: The production and slowing down of neu- 
trons. E. Amaldi: in Encyclopedia of Physics, Vol. XXXVIII/2; S. Flugge (1959). 

[2] Now Comitato Nazionale per PEnergia Nucleare (CNEN). 

[3] See for instance: G. Caglioti: Notiziario CNEN, 11, 44 (1968). 

[4] L. Di Palo, G. Focaccia, E. Lo Prato, F. Marsili, A. Paoletti, M. Perni, D. Pro- 
spers S. Sciuti and A. Verri: Energia Nucleare, 14, 659 (1967). 

[5] M. Giannini, P. Oliva, D. Prosperi and S. Sciuti: Nucl. Phys., 65, 344 (1965). 

[6] R. Cesareo, M. Giannini, P. Oliva, D. Prosperi and M. C. Ramorino: Nucl. Phys., 
A 132, 512 (1969). 

[7] D. Prosperi and S. Sciuti: Suppl. Nuovo Cimento, 5, 1265 (1967). 

[8] G. Ben-David, B. Arad, J. Balderman and Y. Schlesinger: Phys. Rev., 146, 852 
(1966). 

[9] R. Cesareo, M. Giannini, P. Oliva, D. Prosperi and M. C. Ramorino: Results 
presented at the Int. Symp. on Neutron Capture Gamma-Rays, August 11-15, Stud- 
svik (1969). 
[10] For a full bibliography of the work at Casaccia on the subject until 1966, see Noti- 
ziario CNEN, May 1966, p. 85. 
[11] F. Menzinger and A. Paoletti: Phys. Rev., 143, 365 (1966); B. Antonini, F. Men- 
zinger and A. Paoletti: Phys. Lett., 25 A, 372 (1967); B. Antonini, F. Lucari, 
F. Menzinger and A. Paoletti: Phys. Rev., 187, 611 (1969); B. Antonini, M. Felici 
and F. Menzinger: Phys. Lett., 30 A, 310 (1969). 
[12] F. Leoni and C. Natoli: Nuovo Cimento, 55 B, 21 (1968). 



14 



Search for New Stable Particles. 

B. PONTECORVO 

Joint Institute for Nuclear Research - Moscow 



1. - Introduction. 

Everybody knows the extraordinary contribution which Edoardo Amaldi 
has given to the development of modern physics, from X-ray to molecular 
physics, from his classical neutron investigations to problems of nuclear 
techniques etc., not to speak of his great merits in forming young scientists 
and, last but not least, in organizing modern physics in Italy (and not only 
in Italy!). 

As a rule the research work of Amaldi is fundamental and quantitative 
in character, a fact which is evident also in all his first class numerous books 
and review articles, and yet he occasionally likes to perform brave, qualitative 
experiments whose significance is going together with a very small a priori 
probability of finding a positive result (for example, the search for magnetic 
monopoles, or several unpublished old time experiments « a porte chiuse » 
performed in Rome). For this reason I would like to dedicate the present 
paper to my friend Edoardo, with whom my first steps in science are closely 
connected, in the hope that he will not form a too severe judgement of this 
extremely naive piece of fantasy. 

In modern accelerators, to the development of which Amaldi is devoting 
much of his time, the available collision energy is steadily increasing, so that 
the question naturally arises among physicists as to whether there might 
not exist unknown and entirely unpredictable « stable particles » which are 
produced in such accelerators. 

Here stable particles are defined as objects with a mean life >l(h 8 s; 
as it will be explained below, the figure 1(H s is arbitrary and corresponds 
simply to the shortest available pulse of accelerated protons in modern 
machines. 

There has already been performed a number of experiments in order to 
search for new stable particles [1 ]. All the relevant investigations and proposals 



Search for new stable particles 211 

made up till now are characterized by the following circumstances: 

a) the search is made for electrically charged particles, 

b) for the identification of such particles a beam well resolved in 
momentum is analyzed and various quantities (momentum, ionization, time 
of flight, ...) are measured without the decay properties of the new particles 
being investigated. 

Below, a method is proposed for the search of both neutral and charged 
« stable » particles. The advantage of a method which may be operative for 
neutral as well as for charged particles is immediately evident if one keeps 
in mind that among the known particles the number of neutral objects is 
about equal to the number of charged ones. 

In order to discover the new particles it is proposed to study their radio- 
activity properties with the help of a special method. 



2. - The idea of the experiment. 

There are reasons to assume as a working hypothesis that new particles 
with mean lives > 1CH s might exist, that is that the transformation of such 
particles into lighter particles is strongly forbidden in some way. As an 
illustration we could think, for example, that the decay of the new particle 
is due to the second order weak interaction in G, the Fermi constant being 
G = 10- 5 /Mp 2 . Then the probability of decay will be 1/t & G 4 £ 9 , where E 
is a certain energy characteristic of the process. If, for example, the H-hyperon 
had a mass < 1115 MeV, instead of 1315 MeV, its mean life could be longer 
than hours! Besides, the existence of a hyper on with strangeness — 2 and 
mass < 1115 MeV might lead to the appearance of long living quasi-nuclei 
(a sort of hypernuclei) with special properties and, in particular, to a new 
form of radioactivity of matter, in which the decay energy is not measured 
in million electronvolts but is of the order of 100 MeV. However, I wish to 
stress again that this example is only an illustration and the possibility that 
the metastability of the new particles, if they really exist, is to be found out- 
side the boundaries of the known physics seems to me much more plausible. 
Such metastability might be related to the existence of yet unknown quantum 
numbers, or to something else, for example, to an unusual combination of 
known quantum numbers [2]. 

Generally speaking, the body of information accumulated in the region 
of atomic and nuclear physics tells us that metastability is a property appearing 
in the most various phenomena, from phosphorescence to nuclear isomery, 
from the existence of strange particles to the decay K + -^ 7r + +7t°, etc. 



212 B. Pont e cor vo 

I am just proposing to use electronic methods for the search of a new 
type of high energy radioactivity, related to the existence of particles which, 
due to a forbidity of unknown nature, decay with a very long mean life 
(> 10" 8 s). Below, the assumption will be made that these new particles 
are strongly scattered by nucleons. As to the production mechanism of such 
particles, there will not be made any hypothesis. 



3. - How to detect the new particles? 

I shall illustrate here the case when the new particles are electrically neutral. 
Then the discovery of the neutron and of its properties tells us how it is pos- 
sible, in principle, to detect new neutral particles. As it is well known, neu- 
trons may be detected in many ways: 

1) There are detected nuclear recoils in elastic collisions of fast neutrons 
with nuclei (especially protons). Such a method is not adequate for the dis- 
covery of new particles, because their flux is expected to be very small, so that 
the number of nuclear recoils due to the new particles is negligible in com- 
parison with the number of recoils produced by neutrons. 

2) Nuclear reactions produced by fast neutron bombardment with the 
emission, for example, of protons, alpha-particles, etc., are looked for. Such 
a method is also inadequate for detecting new particles, because their flux 
is very small. 

3) There are observed nuclear reactions (n, y), (n, p), (n, a), fission, etc. 
produced by neutrons after they have been slowed down. The possibility 
of slowing down new particles is not to be excluded, but since such particles 
are expected to be generated with an energy of several 10 10 eV, the slowing 
down process requires very large dimensions of moderator (a fact which 
greatly complicates the detection of the new particles, whose intensity is 
very small at best). Under certain circumstances, however, (see below) slowing 
down of new particles could be used. 

4) There are observed radioactive properties of the neutron (generally 
speaking, of the nucleon). Today the observation of the free neutron decay 
is not a difficult problem; however, it is necessary to have a very intense 
neutron beam to observe the decay of free neutrons. The detection of the 
decay of the new particles in their free state is a very unpractical proposition, 
especially if their mean life exceeds 10 -7 s. But the detection of neutrons 
turns out to be quite effective if the decay of bound neutrons (that is if the 
beta radioactivity induced by neutron bombardment) is looked for. The 



Search for new stable particles 213 

analogy for new particles would be the search for a special type of radio- 
activity of pseudonuclei, that is of quasi-nuclei within which the new particle 
is found together with ordinary nucleons (I do not call these quasi-nuclei 
« hypernuclei », because by definition hypernuclei are A quasi-nuclei : hyper- 
nuclei cannot have a mean life much longer than 10~ 10 s). 

It is natural to expect that the new particle (probably produced together 
with other particles) in high energy collisions of protons or y quanta with 
nuclei, as a rule will leave the original nucleus and then will be « stopped », 
either suddenly (after a few collisions) or gradually after slowing down by 
many collisions. For such « stopping » of the new particles a large amount 
of condensed matter is required; I will not discuss here the corresponding 
experiments and I shall note that only radiochemistry, which permits the 
separation of « pure » source of quasi-nuclei from a large amount of irradiated 
material may give positive results (if the lifetime is long enough). 

Below, however, I shall consider the relatively rare but experimentally 
favourable possibility that in a proton or photon collision with a nucleus a 
new particle is produced, which is trapped « at the place of birth » (that is, 
which is found eventually inside the nucleus product of spallation); in such 
a circumstance, a radioactive quasi-nucleus, analogous to a hypernucleus, 
will be produced. Of course this requires that the new particle is being strongly 
scattered (and attracted) by nucleons. Thus the experiment, which will be 
discussed below, consists in the search for a new type of « radioactivity » 
(with mean life > 10 -8 s) in a target, irradiated in a very high energy 
accelerator, the radioactivity being notable for the high energy of its decay 
products (hundreds of million electronvolts instead of million electronvolts 
as in the ordinary radioactivity). 

Immediately there arises the question: What limits on the production 
cross-section of such particles can be obtained from experiments already 
performed ? If the mean life of the new particles is less than a few days, there 
are no limits for the cross-section, because to the best of my knowledge no 
relevant experiments have been performed. Some limits on production cross- 
sections, for mean lives greater than, say a few days, can be obtained from 
the underground experiments of Reines et al. [3] on the degree of accuracy 
with which the baryon conservation law is known. In these experiments it 
was found that the carbon nucleus has a mean life longer than 10 27 y (for 
high energy decays). If we take into account that the carbon compound, 
of which the detector was made, had been irradiated at the earth surface 
by a cosmic ray nucleon flux of 10~ 4 to 10 -5 cm -2 s -1 , the upper limit for 
the production cross-section by nucleons of a radioactive quasi-nucleus turns 
out to be quite large — 10 -30 cm 2 /nucleus. 



214 B. Pontecorvo 

3*1. Possibilities of the method proposed. - Let us discuss now what pos- 
sibilities are given by the method just proposed. An estimate will be made 
for the case of the Serpukhov accelerator, although it is clear that such 
experiments could be performed on an accelerator of the CERN, Brookhaven, 
or SLAC types. Let us consider for example a mean life of the new type 
of radioactivity of the order of days ; in such a case the radioactivity can be 
investigated far away from the accelerator, in conditions of low cosmic ray 
background. In spite of the fact that radiochemical separations will not be 
considered here, still a detection efficiency of about 0.2 or more can be achieved. 
With an average intensity of 10 12 protons/s, at saturation it is possible to 
detect the production of radioactive quasi-nuclei with a cross-section of the or- 
der of 10~ 40 cm 2 /nucleus, which corresponds to about one decay event per day. 
If the production cross-section of quasi-nuclei by protons colliding with nuclei 
is known, one may then obtain the cross-section for the production of new par- 
ticles in nucleon-nucleon collisions after the introduction of a small coefficient. 
It is just the requirement that the new particle is found inside the spallation 
product which leads to the necessity of introducting this small coefficient, the 
value of which, of course, cannot be estimated a priori. However, if we fantas- 
ticate on the analogy between the process considered ihere and the well-known 
process of hypernucleus production, we may give a rough estimate, starting 
from the corresponding experimental data on hypernuclei. It is known that the 
probabilities of hypernucleus production in photoplates by K mesons of energy 
3, 5 and 10 GeV are (3±0.1)% [4], (2.2±0.1)% and (1.2±0.1)% [5] of the 
total nuclear collision probability, respectively. Unfortunately at present there 
are no available data for higher energy kaons, but from the quoted informa- 
tion, and also from the fact [6] that for 25 GeV protons the fraction of nuclear 
interactions in emulsions which results in hypernucleus formation is 0.5%, 
we may guess a value of 0.005 for the indicated small coefficient. 

Thus the proposed method is capable of revealing cross-sections for the 
formation of new particles in nucleon-nucleon collisions which are ten orders 
of magnitude smaller than the total nucleon-nucleon cross-section (of course, 
if the assumptions made are true). 

3'2. Remarks on the proposed method. - If possible, the irradiation of the 
target should require a time comparable with the mean life of the activity 
which is looked for. For short mean lives one should use the extracted par- 
ticle beam (at Serpukhov such a beam will consist of 30 proton pulses the length 
of each pulse lasting 1.5xl0~ 8 s); this permits us also to take the measure- 
ments in the immediate proximity of the target. By means of the classical 
delayed coincidence method (when the radioactivity is looked for in the time 
interval between accelerated proton pulses) one may search for mean lives of 



Search for new stable particles 215 

the order of 10 -8 s with effective beam intensities of a few percent of the 
full beam intensity and of the order of 10~ 6 s or more at full beam intensity. 
When investigating mean lives from 10~ 8 s to a few microseconds one must 
pay attention to the pion and muon background. 

By the way, when searching for the new type of radioactivity with a mean 
life in the microsecond region, the most adequate beam time structure is to 
be found in electron linear accelerators (SLAC and Kharkow), where the beam 
time length is of the order of microseconds with a repetition rate of 100 Hz. 

An extracted proton beam is convenient also when looking for mean 
lives less than a few hours, although in such a case the internal target may 
be used. 

The shortest mean life which can be looked for in the internal target of 
the Serpukhov accelerator is of the order of milliseconds (as such is the time 
required to put the target into the beam). If the internal target is used, it is 
highly desirable to take measurements in one of the straight sections, because 
this allows a larger solid angle to be seen by the detector at the target. 

In the search for activities with mean lives greater than a few hours, the 
internal target can be removed and investigated in conditions of very low 
cosmic ray background and a high solid angle detector. One can consider 
the possibility of using a liquid internal target, which can be easily removed 
from the vacuum chamber. 

In the search for « radioactivities » with long mean lives there are two 
difficulties which are present also, to a less degree, in the search for shorter 
mean lives. 

1) The main source of background is due to cosmic ray muons, the 
integrated flux of which at the earth surface is about 0.01 cm -2 s _1 , and 
also to nuclear « stars » produced by cosmic ray neutrons. It is evident that 
investigating the target « radioactivity » underground has great advantages 
in the search for long mean lives. In the most deep existing underground 
laboratories the cosmic muon intensity decreases by a factor of 10 8 . In such 
conditions there is no background even in the absence of an anticoincidence 
system. Such system, which can easily decrease the muon background by 
a factor of 1000, should be used if the measurements are made near the earth 
surface. 

2) The irradiated target is strongly active due to the presence of spalla- 
tion products. This has the effect that no full advantage for decreasing the 
cosmic ray muon background can be made of the fact that a target of very 
small dimensions (say < 1 cm 2 ) can be used ; as a matter of fact, there will 
be many accidental coincidences between the counters through which pass 
cosmic muons and the small area counter, placed in the immediate proximity 



216 B. Pontecorvo 

of the small target. It may be necessary to place a filter between the target 
and the detector to decrease strongly the beta radioactivity. 

One of the detector elements must be an energy spectrometer, let us say 
a Nal chrystal (or a lead glass spectrometer, etc. if high energy gamma's 
are looked for). 

If the measurements are made at the earth surface it may turn out to be 
necessary to use some kind of track chamber to reject the events in which 
the particles are not coming out of the (small) target. 

Here I would like to mention another possible registration arrangement. 
When a heavy (Z>80) quasi-nucleus decays, the decay products may in- 
duce with reasonable probability the fission of the nucleus. Consequently 
there raises the probability of searching for a « radioactivity » with emission 
of fission fragments in a thin heavy target (made of an element not under- 
going spontaneous fission, let us say Th) irradiated by high-energy particles. 
The interest in this arrangement is due to the possibility of detecting (even 
at the earth surface) very rare fission events of a substance having an ex- 
tremely high beta activity. 

One might also consider the search, deep underground, for a delayed emis- 
sion of a few neutrons from a heavy material irradiated by high-energy par- 
ticles, because it is well known that a heavy nucleus excited at a few hundred 
million electronvolts emits many evaporation neutrons. 

4. - Conclusion. 

The well-known methods of observing neutral particles (decay in fight, 
missing mass spectrometer) are adequate only if the mean life is short enough 
or if the corresponding production cross-section is relatively large. 

It is evident that the present proposal (a search for a « radioactivity » of 
a special type) is quite naive, a fact which I fully recognize. However the 
proposal is relatively simple and, independent of the ideas expressed in this 
paper, the suggested experiment has a definite phenomenological interest. 



It is a pleasure for me to thank R. Vassilkov, L. Landsberg, L. Okun, 
M. Markov, L. Nemionov, A. Ciudakov for support and discussions. 



Note added in proof. - After this paper was written, Dr. Giacomelli has kindly informed 
me about an interesting investigation [7], which is relevant to the question discussed above 
from an experimental point of view, although it originated from a completely different 
« phylosophy ». A search was made for magnetic monopoles, which might have been 



Search for new stable particles 217 

produced in collisions of high-energy protons with nuclei. In order to detect the products 
of a possible monopole-antimonopole annihilation, the authors lookedfor a high energy 
radiation from a target irradiated by 27.5 GeV protons. No effect was found, the detector 
being sensible to electrons and photons in the time interval from 0.1 s to 1 day after 
the « production of the monopole-antimonopole pair » in targets of Al, polyethilene, and Cu. 
According to this investigation the upper limit for the production cross-section in light 
elements of a radioactive quasi-nucleus of the type discussed in this paper turns out to 
be several orders of magnitude smaller than that from ref. [3]. 



REFERENCES 



[1] See for example, Antipov et al.: Phys. Lett., 29 B, 245 (1969). 

[2] See for example, L. Okun: JETP, 47, 1777 (1964). 

[3] H. Gurr, W. Rropp, F. Reines and B. Meyer: Phys. Rev., 158, 1321 (1967). 

[4] J. Lemonne et al.: Nuovo Cimento, 41, 235 (1966). 

[5] G. Coremans et al.: Nuovo Cimento, 61, 525 (1969). 

[6] J. Zakrzewski: Proc. Intern. Conf. Hyper fragments (Geneva, 1964). 

[7] M. Fidecaro, G. Finocchiaro and G. Giacomelli: Nuovo Cimento, 22, 657 (1961); 
E. Amaldi: On the Dirac Magnetic Poles, Old and New Problems in Elementary Par- 
ticles, edited by G. Puppi (Academic Press, New York, 1968). 



The Isobaric Analog Resonances 
in Phenomenological Nuclear Spectroscopy. 

R. A. Ricci 

Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Legnaro - Padova 
Istituto di Fisica G. Galilei, Universitd di Padova - Padova 



1. - Introduction. 

The remarkable achievements of nuclear spectroscopy in the last ten 
years have been made possible by the advent of new types of high resolution 
accelerators, such as Tandem Van de Graaff, to produce particle beams in 
a large interval of masses and energies, and by the solid state detectors upon 
which the so-called « in beam nuclear spectroscopy » is based. These tech- 
nical advances have allowed measurements that only a few years ago were 
impossible and a considerable volume of new information is now available 
for theoretical investigations. A large number of nuclear states have been 
identified and classified by means of precise determinations of energy, an- 
gular momentum, parity, and transition probability. A systematic knowledge 
of such attributes of nuclear levels is essential for any attempt to relate them 
theoretically and provide the experimental ground for testing the validity 
of the current microscopic theories the goal of which is to disclose the fun- 
damental features of the nuclear structure. 

The aim of this paper is to give a comprehensive outline of the so-called 
isobaric analog resonances, a promising and relatively new topic in nuclear 
physics. The outstanding importance of the isobaric analog resonances 
(IAR) is brought about by the fact that in many cases they give the clue for 
the correct interpretation of the observed structures in nuclear excitation 
spectra. 

It is well known that the low excitation levels (bound states) are charac- 
teristic of the collective motion of the nucleus, namely rotation and/or vi- 
bration modes or motion of mutually interacting particles. This situation 
corresponds to an average width r (decay probability) very small as com- 



Isobaric analog resonances 219 

pared to the average distance D between levels, i.e., J 1 <C D. By gradually 
increasing the excitation energy, the level density also increases and, above 
the nucleon threshold, the nucleus « enters » the continuum region. In the 
neighborhood of this region the nuclear states, obtained through capture 
reactions (formation of the compound nucleus), decay to the levels of the 
residual nucleus and the yield relative to the different exit channels exhibits 
a resonant behavior (compound nucleus resonance) : Since r is of the order 
of D (r~D) the resonances are observed separated provided the measure- 
ments are carried out with adequate energy resolution (§£'</'). 

At higher excitation energies the situation becomes more complicated 
due to the overlap of the nuclear levels. The existence of statistical fluctua- 
tions, namely a sort of background noise due to the behavior of the reac- 
tion amplitudes, is nowadays a well-established fact. The reaction partial 
amplitudes, randomly distributed over a large number of levels, sum up 
coherently with a width r c (coherence width): this implies that when 8E<T, 
namely when the measurements are performed with high resolution, the fluc- 
tuations can be observed as superimposed to an average behavior generally 
nonresonant-like [1]. Characteristic resonances, however, can distort the 
nonresonant behavior: evidence has been obtained in the last few years of 
such an anomalous behaviour in experiments with not very high energy res- 
olution. This kind of resonances has been called « intermediate structures »: 
They correspond to peculiar nuclear states, completely determined by the 
entrance channel, which give rise to the excitation of resonances of the com- 
pound nucleus through particular correlations (doorway states) [2]. The im- 
provement of the resolution brings to view a fine structure composed by 
several narrow resonances which should be properly correlated in order to 
give the single gross resonance observed in experiments which de facto aver- 
age the reaction cross-section. It is evident that a clear-cut identification of 
an intermediate structure is bound to the fact that the observed effect on 
the cross-section be unambiguously independent of any averaging procedure. 

It is conceivable that by virtue of some specific interaction the doorway 
state mixes coherently in the resonant region with the compound nucleus 
resonances nearby localized. The stronger such a mixing is, the more cor- 
related are the nuclear states with the entrance channel, contributing to the 
intermediate resonance and more crucial is the effect of the presumptive 
doorway state on the decay behavior of the compound nucleus. So far these 
predictions have not yet been corroborated by a clear-cut experimental sup- 
port because of the difficulty of performing measurements satisfying the fol- 
lowing conditions [3]: 

D z >r It D z ^D,2r, 



220 R. A. Ricci 



where Z> 7 is the average spacing of intermediate states and .T z is their average 
width, while D and r refer to the compound nucleus. There are two cases 
where the above conditions are fulfilled : the giant resonance and the isobaric 
analog resonance. 



2. - Isobaric analog states as resonances. 

The essential standpoint was the discovery of the possibility of producing 
isobaric analog states as compound-nucleus resonances in reactions initiated 
by low-energy protons (below the Coulomb barrier) on medium and heavy 
nuclei. The pioneer experiment was performed by the Florida State group [4] 
who found strong anomalies in the excitation curve of the 89 Y(p, n) reaction 
clearly arising from the 89 Y+p compound system ( 90 Zr) and reproducing with 
good accuracy the intervals of the low-lying bound states of the 89 Y+n ( 90 Y) 
isobar. 

The standard analysis of the elastic proton-scattering data revealed the 
exact correspondence of spin and parities ; moreover the « analog spectrum » 
was found to be displaced relatively to the isobaric « parent spectrum » by 
the expected Coulomb energy difference. This was the second important 
step in the modern history of the isobaric-spin nuclear spectroscopy. The 
first one was the identification of new modes of excitation in charge-exchange 
reactions as isobaric analog states. The anomalous sharp peak found in the 
neutron evaporation spectrum following the 51 V(p, n) 51 Cr reaction by Ander- 
son and Wong [5] was interpreted as the isobaric analog of the target ground 
state. Since then this situation was found to hold for other heavier nuclei 
and not to be restricted to light nuclei as it was believed for a long time. 

These experiments and their interpretation in terms of isospin dependence 
of the optical potential, as given by Lane [6], opened the current interest 
in isobaric analog states (IAS) and buried the old superstition that the clas- 
sification of nuclear states in terms of isobaric spin was useless for medium 
and heavy nuclei (i.e., Z> 10) because of the strong Coulomb interaction. 
In this connection it is worthwhile to mention that since 1961 French and 
Mac Farlane [7] have introduced the concept of isobaric splitting in describing 
the distribution of spectroscopic strengths in particle-transfer reactions even 
in medium-weight nuclei. 

In fact, the use of isobaric spin in nuclear spectroscopy since the theory 
developed by Wigner in 1937 [8], was strictly connected with the occurrence 
of isobaric-spin multiplets in the assumed charge-independent nuclear world. 
In this world the knowledge of the properties of one member of a given 
multiplet is sufficient to describe completely the other members (the « isobaric 



Isobaric analog resonances 221 

analogs » of the former). However, the nuclear Hamiltonian is not exactly 
charge-independent; in the limits that charge-dependent effects such as the 
Coulomb interaction between protons are not so strong to cancel the regu- 
larities predicted by the charge-independent part, the isobaric spin is still a 
good quantum number. In light nuclei the Coulomb mixing is only a very 
slight perturbation because of the quite large average spacing between states 
of different isospin ; consequently, isobaric multiplets were found to hold in 
this case with analog states obtained by simply applying the charge-exchange 
operator T~ to the parent states. 

The situation is different for medium and heavy nuclei, where the sym- 
metry and pairing energies can put the analog states in the continuum region 
and the strong Coulomb interaction can mix them with nearby states of 
different isospin ; moreover they can become proton unstable, while the parent 
states (low-lying) are stable against particle emission (bound). The con- 
nection between such states cannot be simply assured by the T~ operator and 
the isobaric-spin correspondence may be completely destroyed. 

The discovery of isobaric analog resonances (IAR) has been taken as an 
indication of the survival of the isobaric-spin (charge-independent) description 
in the continuum region, where the IAS can be interpreted as some kind 
of « special states » embedded in a dense spectrum of complicated com- 
pound-nucleus states [9]. 

A schematic picture of the correspondence between such IAS and parent 
states is shown in Fig. 1. 

The (Z — 1, N+l) and (Z, N) nuclei are connected through the charge- 
exchange operation (n^p); neglecting the Coulomb interaction, it may 
happen that the levels of the former ones are located at exactly the same 
energy in the continuum region of the latter ones. Due to symmetry effects 
in the nuclear interaction, the displacement of the states r> and T< is ex- 
pressed by the term £ sym the value of which for medium and heavy nuclei 
may be higher than the threshold for neutron emission. Without Coulomb 
interaction the states T> in the (Z, N) nuclei would have exactly the same 
configuration of the parent states in the (Z— 1, jV+1) nuclei; because of 
the isospin difference, these states do not mix with those of the continuum 
and therefore are bound states. 

The Coulomb interaction gives rise to a displacement of the spectrum of 
the (Z, N) nucleus which amounts to the difference A£c (*) between the 
Coulomb energy of the two systems; furthermore, the proton emission 



(*) The term AE C considered here accounts also for the neutron-proton mass dif- 
ference which should be included in the atomic mass scale. Only relative shifts are 
considered here. 



222 R. A. Ricci 



with Coulomb without 
interaction Coulomb int. 



analogue states 




Z^N 



T=T.= 



N-Z 



Fig. 1. - Correspondence between isobaric analog and parent states with and 
without Coulomb interaction in a typical medium or heavy nucleus. 



threshold is lowered and it may occur that the analog states become unstable 
against proton emission. In this case it is possible to excite an analog state 
as a resonance by means of elastic scattering of protons. It has to be re- 
marked that the Coulomb interaction favors the mixing of the T> state with 
jhe neighboring ones correlated with it by the same set of spin and parity 



Isobaric analog resonances 



223 



attributes. In this way, exit channels responsible for the decay of the T< 
states are open; it follows that, by virtue of charge-dependent effects brought 
about by the Coulomb mixing, an analog state becomes indeed observable 
through an isospin-forbidden decay (for example, neutron emission). In the 
first experiment carried out by the research group of the Florida State Uni- 
versity [4], which led to the discovery of the isobaric analog resonances, 
anomalous nuclear structures were observed in the excitation curve of a 
(p, n) reaction. 

In conclusion, an analog state in the continuum appears as an inter- 
mediate structure induced by an elementary excitation, namely a charge- 
exchange interaction, which stimulates the excitation of the neighboring 
states of the compound nucleus through the Coulomb mixing and acquires 
their decay properties. 



3. - Production and decay modes of IAR. 

Figure 2 shows a schematic diagram of a proton-induced reaction with 
excitation of IAR and the isospin-allowed and forbidden decay of these latter. 



,t E P 



(Z-1.N) 
target nucleus 




nuclei 



(Z,N) 
T 2 -NzZ 



(Z-1,N+1) 

parent nucleus 



■z- 2 
Compound nuctous 



Fig. 2. - Schematic diagram of a proton-induced reaction with production of 

IAR; the neutron or a-decay of IAS is made possible by the coupling with 

the normal T < compound nucleus states. 



224 R. A. Ricci 

The bombarding proton energy (in the center-of-mass) needed for ex- 
citing IAS via a resonance reaction is: 

£c.m. = A£ c — Sn , 

where A£c is the Coulomb shift and S a the neutron separation energy. 

In light nuclei, in spite of the low Coulomb barrier, states which are the 
analog of low-lying bound states cannot be excited in resonance reactions, 
since &E C < S a . Now, AEc increases with Z while S n decreases slowly with 
increasing A; consequently IAR become available for medium and heavy 
nuclei. On the other hand the required proton bombarding energy becomes 
higher and higher; this explains the fact that such experiments are generally 
done with Tandem accelerators which cover the energy interval required 
for a large amount of nuclei and whose energy resolution is far superior 
to the energy resolution of either the incident proton or the outgoing par- 
ticle detection devices. 

Nevertheless, in recent years, some interesting investigations on IAR 
have been performed even at subtandem energies (up to 5.5 MeV) with Van 
der Graaff accelerators in connection with properly used proton and gamma 
high-resolution spectrometry. In fact, the available experimental data on the 
fine-structure and gamma-decay properties of IAR have been obtained mostly 
at subtandem energies and in the region of medium- weight nuclei (2s-ld 
and 1/j shell). 

These properties are of fundamental importance in the analysis of IAR 
and the related spectroscopic information. Furthermore, the standard elastic 
scattering analysis in terms of interference pattern between resonance and 
potential scattering allows the extraction of spectroscopic parameters (reduced 
proton widths) related to the properties of the parent bound states. 

3'1. Elastic scattering and spectroscopic information. - An IAS in the 
continuum is a resonance. A fundamental problem is to explain how it can 
be putted in the same isobaric multiplet of a bound state. More precisely, 
the fundamental task is to understand the connection existing between the 
appearance of an isobaric analog resonance and the strict isobaric corre- 
spondence with a bound state and what role is played by correlations, dif- 
ferent from the simple Coulomb mixing, between the entrance channel and 
the normal compound nucleus state in generating the observed resonance. 
Several theoretical approaches attempt to find an answer for all these open 
problems and to develop reliable methods for the analysis of excitation curves 
with the final goal of extracting the spectroscopic parameters characterizing 
the IAR [9, 10,11]. 



Isobaric analog resonances 



225 



From a phenomenological point of view, which is of interest here, the 
IAR observed in proton elastic scattering can be related directly with the 
corresponding parent states via their spectroscopic properties. A typical 
curve for a (p, p) elastic excitation function is shown in Fig. 3. 

The curve is analyzed in terms of a resonance plus potential scattering 
yielding to the resonance energy E R , the total width r, and the partial pro- 
ton width r p ; moreover, the interference pattern observed at different scat- 





"Zrlp.pJ^Zr 


0) 

<** 

'c 


f ^K 165° 








\J Ep crn ( MeV ) 

i i i 



300 - 



200 - 



100 



5.8 



6.0 



mb 



6.2 



92 Zr(p,n) 92 Nb 




cm 



Ep" m (MeV) 



5.7 



5.8 



5.9 



6.0 



6.1 



6.2 



63 



Fig. 3. - The excitation function for elastic proton scattering on 92 Zr in the 
region of the 93 Nb ground-state isobaric analog (schematic, see ref. [12]). Also 
shown is the neutron excitation curve for the 92 Zr(p, n) reaction in the same 

energy range. 



15 



226 R. A. Ricci 

tering angles yields to the proton angular momentum / p and consequently 
to the parity, and, in some cases, to the spin of IAR. The spin is uniquely 
determined when also the polarization of the scattered protons could be 
measured [13]. The correspondence with the isobaric parent state is first 
given by energy and quantum numbers. Furthermore, a clear correlation 
exists between the elastic width and the stripping spectroscopic factors of 
these states. 

From Fig. 2 it is seen that the low excitation states of the parent isobar 
(T '—T z = T>) may be produced by means of a stripping reaction through the 
transfer of one neutron to the same target nucleus. The spectroscopic infor- 
mation thus obtained concerns not only the energy, spin, and parity but also 
the spectroscopic factor Sd.p, which, in single-particle reduced width y| >p 
units, expresses the reduced width y\ for neutron capture into a bound parent 
state: S d>p = yljy% . 

Starting from the same target state the elastic scattering cross-section for 
the analoge resonance yields to the reduced width y\ for proton capture 
into the analog state, through the relation y% =r^ bs /2P l , where r° bs is the 
observed partial width and P z is the trasmission coefficient which allows the 
proton to traverse the momentum discontinuity at nuclear surface and to 
penetrate the angular momentum and Coulomb barrier. 

If T is the isobaric spin of the target the isobaric analog state spends only a 
1/(2T" +1) fraction of the time as a proton state; consequently: 

yl i 



yl 2T +i' 

i.e., 

y* p 2r +r d ' p - 

This result gives the correspondence between the nuclear structure of the 
analog resonance and that of the parent state through the correlation with 
the same initial state (target nucleus). Unfortunately, current methods used 
for extracting the spectroscopic factors by means of distorted wave analysis 
give results affected by large uncertainties (of the order of (20^-30)%), 
whereas the determination of the reduced width from elastic scattering ex- 
periments is generally limited because of difficulties in evaluating correctly 
the trasmission coefficient which may be modified in a complicated way by 
the presence of the !T< states. The study of the IAR will become an independent 
spectroscopic tool of outstanding importance as soon as appropriate methods 
will be developed for the determination of accurate spectroscopic factor 
from the data [9]. So far the study of the inelastic scattering through the 



Isobaric analog resonances 227 

IAR seems to be more promising. In this case, the excited states of the target 
nucleus are correlated with the isobaric analog resonance. The comparison 
of the inelastic and elastic excitation curves gives information on the config- 
urations existing in those states. 

3*2. Fine structure. - The « gross anomalies » observed in proton elastic 
scattering have generally total widths of the order of 10 times the widths 
expected for the normal compound-nucleus resonances. This is what is found 
in poor-resolution experiments and is generally interpreted in terms of a 
« gross structure » averaging over individual resonances just as an inter- 
mediate state. The main effect of the 7"> resonance is associated with the 
interference between Coulomb and resonance scattering. However, the shape 
of the gross anomaly shows some departures from the pure Breit-Wigner 
form, with a typical asymmetric behavior (see Fig. 3). 

This was interpreted by Robson [9] in the framework of the i?-matrix 
theory as a consequence of a coherent interference between two contribu- 
tions arising from two different regions of the configuration space: one is 
internal to the nucleon-charge distribution (r < re), where T< is a good 
quantum number (no Coulomb mixing); the other is the external region 
allowing Coulomb mixing. When the two solutions of the Schrodinger equa- 
tion are smoothly joined in order to calculate the collision matrix, the presence 
of the T> IAS contributes to the widths of the normal compound-nucleus 
states with the same attributes defined in the internal region, giving rise to 
an enhancement of such states. This enhancement is asymmetric as a function 
of the energy and vanishes just at the energy of the IAS, which is found to be 
higher than the energy at which the elastic resonance occurs. 

This prediction was confirmed in the high-resolution experiments where 
the fine structure of IAR resolved. The typical example is shown in Fig. 4, 
where the fine structure covered by the IAR found in the 92 Mo(p, p) exper- 
iment is seen to quench more rapidly on the high-energy side of the excita- 
tion curve. 

We are dealing, in this case, with a typical heavy compound nucleus ( 93 Tc) 
with high normal level density in the analog state region; then the fine struc- 
ture is interpreted as being due to the fluctuations of many overlapping levels 
with the same spin and parity as the analog one (r>) but with lower isospin (T<). 
This is related to the fact that the best fit to the experimental cross-section, 
in the poor resolution measurement, yields a partial proton width r p which 
accounts for only ~ \ of the total width r B ; r p = r R would have been ex- 
pected in this case, since we are below the Coulomb barrier and the sole 
proton channel is open [15]. More insight in this matter is gained when the 
experiment can resolve all the T< individual components in the region of the 



228 R. A. Ricci 



E x (MeV) 


dc/dXl 

1 300 

Q(p.n) 


r rir 

- Ep cm « 5.24 MeV 


250 

Mo + n 


- 92 Mo + p \ 

- ae c «i; 

: 1 


2.28 MeV 25 

U9 

20 

93 V2+ 

A2 Mo 51 
T=9/2 


93 Tc V 
T< = 7/2 



92' 



90° 



_i i i i_ 



T- 0.25 keV 




Ep(MeV) 



528 



5.30 



532 



Fig. 4. - Gross and fine structure of the IAR corresponding to the 93 Mo ground 

state (T=T Z = %) in the 92 Mo(p, p) reaction (schematic, see ref. [15]). On 

the left-hand side the schematic diagram of the reaction is shown. 



analog state. Few experiments of this type have been performed so far, all 
confined to medium-weight nuclei such as 41 K [16], 49 Sc [17-20], 23 Na [21], 
43 - 45 Sc [22]. It is interesting to note that, due to the rather low neutron sep- 
aration energy in all these cases such experiments have been performed at 
subtandem energies (i.e., with the Van de Graaff accelerators of the Duke, 
Padua and Utrecht Universities). The high overall resolution obtained in 
these experiments (from 0.2 to 0.8 keV) could separate the individual com- 
pound-nucleus resonances averaged over by the IAR. In fact we are dealing 
here with nuclei where the level density at the IAS energy is not so high as 
in heavier nuclei, so that the best energy resolution actually available from 
accelerating machines is enough for such measurements. 

All these experiments clearly show that the sum of the partial widths r pX 
is less than the total width r R of the intermediate resonance, i.e., the tails of 
the individual resonances add coherently to produce one large resonance 
with a great amount of nonresonant range inbetween. The difference 
W=T R — ^r pX is the so-called « spreading width » (statistical contribution 

in the region of the IAS). 

Figures 5 and 6 show schematically two typical examples of fine structure 
experiments, the first one being the 40 A+p reaction with essentially the proton 



Isobaric analog resonances 



229 



channel open [16], the second being the 48 Ca+p experiment where also the 
neutron and gamma-ray channels are open and indeed measured [17-19]. 

In the first case 17 individual resonances were found, every one with spin 
and parity f _ and with a distribution corresponding exactly to the shape 
of the Lane-Wigner-Thomas giant resonance in 41 K; this « micro-giant » 
resonant structure corresponding to the analog strength being shared among 
the surrounding normal states, was already predicted by Bloch and Schiffer [23], 
for the case where the spacing between such states is much less than the mixing 
strength. The enhancement of the compound-nucleus resonance appears clearly 
as due to the mixing with the nearby analog state, while the fact that, for 
each resonance, r p <T shows the presence of a nonvanishing spreading 
(damping) width, arising from the coupling with other nonresonant degrees 
of freedom of the target plus proton system [11]. 

In the second case the IAR corresponding to the 49 Ca ground-state isobaric 
analog in the 48 Ca+p ( 49 Sc) system was resolved in individual components 
singled out in the isospin-forbidden (p, ny) reaction [17] performed with 
the 5.5 MeV Van de Graaff of Padua indicating a more detailed structure 
than that found in a first (p, p) experiment with not very high energy resolu- 
tion [19]; a successive high-resolution elastic scattering experiment performed 
at the Utrecht Van de Graaff accelerator [18] confirmed the presence of at 



10 



r 5 

Uj 



E - 1.825 MeV 
p 

/_ 



A + p 



a) 




T =5/2 



r = 800 eV r= 1250 eV 



j — i — i i i i i_ 



.86 1.87 

E (MeV) 



- T 



3/2 



Fig. 5. - Reaction diagram (a) and fine structure (c) of the 1.87 MeV IAR found 
in 40 A(p, p) experiment at Duke University [16] (schematic); the IAS (7^ = f) 
corresponds to the fourth excited state (T = f , / = f") of the parent 41 A. The 
corresponding gross structure found earlier at Yowa [12] is also shown sche- 
matically (b). 



230 



R. A. Ricci 



Ex(MeV) 



10 - 



T>= 9/2 



48 



Ca+p 




J 
Ep c =1.935 Me' 



LQJB.QJT. ^Sc 



AE r = 7.081 MeV 



&■ 



49 Ca 
T=9/2 



46 



Ca+n 



49 



i/i- 



Sc 

21 3t 28 

T<=7/2 




8x10 3 A ^rp-1.5keV 

3/2- 172 « 
6 




1.95 1.96 1.97 1.98 1.99 200 
«labu 



Ep iaQ (MeV| 



Fig. 6. - Reaction diagram for the excitation and decay of the 49 Ca ground- 
state IAR in the 48 Ca+p system, and corresponding fine structure found in 
(p, p) and (p, ny) experiments (refs. [17] and [18]). 



least 7individual resonances with the expected f assignment and with 2^p* = 

= (2.2±0.3) keV rather smaller than the single-particle value 4.4 keV obtained 
from the stripping spectroscopic factor Sd, P = 1 found in the 48 Ca(d, p) 49 Ca 
reaction [24]. A similar result has been obtained in the recent experiment 
performed at Duke [20], where an additional §~ resonance has been found 
and a value of 1.85 keV for the sum of the proton widths has been reported; 
this value, compared with the estimated [19] width for a single-particle (f - ) 
resonance, gives a spectroscopic factor .Sd.p = 0.6. 

3'3. Shell model description of IAR. - The 48 Ca+p system is interesting 
from several points of view. First, it corresponds to a particle outside a doubly- 
closed shell core ( 48 Ca has Z = 20 and N = 28) ; consequently it is a case 
suitable for a shell-model description of the isobaric analog resonances [10, 25]. 



Isobaric analog resonances 



231 



The basis for a microscopic shell model analysis of IAR is the particle-hole 
picture of Bloch and Feshbach [2], where the nucleon-nucleon interaction 
allows the mixing between the virtual state of the proton plus target system 
with a 2 particle- 1 hole state in the compound nucleus; such 2p-\h state is 
the doorway state with isobaric analog configuration (T = T>), giving rise 
to the intermediate structure in the presence of the neighboring T< states; 
these latter correspond to much more complicated configurations (3p-2h, etc.). 
The particle-hole picture for the 49 Ca ground-state analog is schematically 
represented in Fig. 7. 



•hell p ! n 



48. 



W ( Ca ) 

ANALOGUE T, - T + 1/2 - 9/2 




2s-1d 



n-p 



• 






1 


I 


s, 





<M 




W, 


p 


§ 




PARENT 

49 Ca 
T z -To+V*-9/2 



Z N 

1 p state 



2 T ( 2 p - 1 h ) states 
COMPOUND : 48 Ca+p 

Tr-To-V2-7/2 



Fig. 7. - Particle-hole picture of the 49 Ca ground-state analog configuration. 
The 2p-\h configuration corresponds to one neutron in the 2/?| shell and one 
proton and one neutron-hole in the lf^ shell; since there are 8 neutrons in the 
1/? shell, there are 2T modes of obtaining the 2 particle - 1 hole doorway state 
(T Q = 4 is the target isospin). 



The 2p-lh state serves as an intermediary between the incident proton 
and the complicated 3p-lh states, since it can mix with them while the proton 
entrance channel cannot; the doorway can decay into the T< states forming 
an intermediate structure, whose total width is given by 



r R = r\+r\, 



where r^ is the « escape width » for the single-particle decay in the con- 
tinuum and ri is the « damping width » (corresponding to the spreading 
width already mentioned) i.e., the decay width into more complicated modes. 
Now r"t = ^ r t , where the index / refers to all the open exit channels (i.e., p, n 



232 R. A. Ricci 

and y in our case), consequently 

Following the results of refs. [17-19] and the analysis of the intermediate 
structure performed in ref. [19], one has 

r p + T„ = (2.4 ± 0.5) keV (refs. [17, 18]) = (3.4 ± 0.8) keV (ref. [19]) , 

T Y =0.24eV (ref. [18]), 

r R = (4.7 ± 0.4) keV (ref. [19]) , 

r exp =(2.3 ±0.7) or (1.3±0.8)keV, 

theory = 0.6 keV (ref. [19]). 

A better evaluation of the widths for the isospin-forbidden neutron decay 
seems to be necessary in order to ascertain if there is a real discrepancy 
between the experimental and calculated damping width; this is of some 
importance since a large spreading width corresponds to the interference of 
other possible degrees of freedom to be considered in a suitable classifica- 
tion [11]. 

3"4. Neutron decay. - A second interesting aspect of the 48 Ca+p exper- 
iment is the presence of the neutron exit channel. Since it is generally assumed 
that the low-lying levels of the residual nucleus arising from the (p, n) reac- 
tion ( 48 Sc in the present case) are normal T< = T z = T — 1 states, with 
reasonably pure isobaric spin, the neutron decay from the r> = T + ^ 
IAS should be forbidden (Ar = f ) (see Figs. 2 and 6). As already men- 
tioned, its occurrence is taken as an indication of the mixing between the IAS 
and the surrounding T< states ; on the other hand this makes it possible to 
characterise the IAR through the sharing of its strength among the normal 
compound-nucleus states. The case in point here is a typical example. 

Indeed it was found in a first poor resolution experiment (see ref. [17]) 
that the neutron decay at a proton bombarding energy corresponding to the 
gross resonance was selective for the residual states of 48 Sc. In other words, 
there exists a preferential decay of the analog state to one of the levels of the 
residual nucleus (due to a particular configuration of the latter one), namely 
to the level at 1 .4 MeV which undergoes a strong gamma transition of 
0.78 MeV. The yield of this transition, determined by means of gamma 
spectroscopy, provides an excellent measurement of the excitation of the 
analog resonance and of its fine structure. This is clearly seen in Fig. 8, 
where the yields of 0.78 MeV and 0.37 MeV Y-rays are compared, the latter 



Isobaric analog resonances 233 



48 



Ca(p.nj) Ej= 0.78 MeV 



JU 




Ej= 0.37 MeV 




2.K) 2]2 
Ep ( MeV ) 

Fig. 8. - Yields of the 0.78 and 0.37 MeV y-rays following the neutron decay 
of the 49 Sc compound-nucleus in the region of the T = § , / = f IAR, in the 



48 Ca(p, ny) experiment, a) E y 



0.78 MeV; b) E y = 0.37 MeV. 



one being originated in transitions from a lower level which collects most of 
the neutron decay from the compound nucleus states. The selection of the 
0.78 MeV y-ray allows, therefore, the suppression of the background due to 
resonances and decay modes different from those mixed with the analog 
state. 

Another interesting fact is that the shape of the enveloping distribution 
is almost Gaussian, as is expected in as much as the entrance channel can 
only produce the higher isospin and the neutron emission can only proceed 
from the lower isospin; the asymmetry of such a distribution is less pronounced 
than in other typical cases, where a dip is found in the neutron excitation 
curve just at the energy of the IAS at which the isospin-violating processes 
should vanish following the Lane-Robson model (see Fig. 3). 

3'5. Gamma (radiative)-decay. - The third point of interest in the present 
example is the determination of the gamma-decay properties of the IAR. 
Such properties are of considerable interest in nuclear spectroscopy, since the 
gamma-transition probabilities are strongly correlated with the intrinsic struc- 
ture of the nuclear states. In fact it has been pointed out that the gamma- 
decay of IAS observed in (p, y) reactions may be very simple [26]. Let us 
consider the case of quite pure IAS (r> = 7^+1) with single-particle 
character in the compound nucleus formed in a (p, y) reaction ; in the same 



234 



R. A. Ricci 



T>= 9/2 = Tp + 1/2 



Compound nucleus 



T>=T +1/2 





J = 3/2 



Sd . p f f rom A8 Sc )- 0.6 \ t T<=7/2 = T - V2 
* J« 3/2 



W/2 



J-7/2 



""Sc 

21 28 



Fig. 9. - Schematic picture of isospin splitting of a single-particle state as a 

result of its coupling with a J = core. The 48 Ca+p case is illustrated with 

the result found in (p, y) reaction (ref. [18]). 



nucleus at least 1 bound state with exactly the same nuclear configuration, 
but with different isospin symmetry (T< = T z ) exists, due to the isospin 
splitting of single-particle states [7] (see Fig. 9). Such a state is called an 
« antianalog » [26] or « homolog » [27] state and should be connected with 
the IAS via a strong Ml transition [28] of the order of the Weisskopf unit. 
The work of the Utrecht [26] and Ohio [28] groups in the 2s-ld shell was 
successful in locating, through (p, y) reactions, several strong IAR with rather 
little isospin mixing and strong Ml analog to antianalog transitions. Such 
cases are rather simple, because only the channels for proton and gamma-ray 
emission are open, the analog and antianalog states are essentially not frag- 
mented and the corresponding gamma-transition connecting them stands out 
clearly in the gamma-spectrum taken at the resonance energy. 

This is due to the weak coupling between the single-particle state and the 
core which forms a (7 , <, 7^) doublet with spin J = j if the core is a J = 
nucleus (see Fig. 9). Theoretical calculations performed by Maripuu [29] 
show that a strong enhancement of Ml transition rates between the two 
members of the isospin-doublet arises for the « parallel » case J = j = / + | 
(i.e., when the orbit and the spin of the added proton are parallel) as com- 
pared with the « antiparallel » case J=l—%. The experimental data in the 



Isobar ic analog resonances 235 

2s-\d shell nuclei generally agree with these predictions. A rather different 
situation is found in the 1/j shell, where data concerning 49 Sc (we are discussing 
here), 49 V [30] and 51 V [31] are now available. 

In the 49 Sc case, for instance, the coupling between the \p§ single-particle 
state and the 48 Ca core seems not to be weak so that it becomes fragmented, 
especially for the lower (r<) member; moreover, as we have seen, the IAS 
is rather strongly mixed with the neighboring compound-nucleus states with 
complicated configurations. The gamma-branching from the IAR is then 
less simple, but could give important information for a microscopic description. 

The result found in the 48 Ca(p, y) experiment [18] show the presence of 
an E2 ground-state transition from the major resonance present in the fine 
structure of the IAR and the absence of Ml transition (less than 6 x 10~ 4 
Weisskopf unities) to the expscted strongest fragment of the antianalog 
state (the 3.08 MeV level with stripping spectroscopic factor of about 0.6). 
This could be related to a much more fragmentation of the 1/7$ single-particle 
state and to a strong mixing of the IAS with more complicated (3p-lh) con- 
figuration at such a resonance energy. 



I would like to express my gratitude to my friend Professor C. Villi for 
critical reading of the manuscript. 



REFERENCES 

[1] See for instance, T. Ericson and T. Mayer-Kuckuk : Ann. Rev. Nucl. Sci., 16 (1966). 

[2] See for instance, B. Bloch and H. Feshbach: Ann. Phys., 23, 47 (1963). 

[3] Cfr. C. Bloch: Intern. Conf. Progress Nucl. Phys. with Tandems, Heidelberg, July 1966. 

[4] J. D. Fox, C. F. Moore and D. Robson: Phys. Rev. Lett., 12, 198 (1964). 

[5] J. D. Anderson and C. Wong: Phys. Rev. Lett., 1, 250 (1961). 

[6] A. M. Lane: Nucl. Phys., 35, 676 (1962). 

[7] J. B. French and M. H. MacFarlane: Nucl. Phys., 26, 168 (1961). 

[8] E. P. Wigner: Phys. Rev., 51, 106 (1937). 

[9] D. Robson: Phys. Rev., 137, B 535 (1965); Intern. Conf. Progress Nucl. Phys. with 

Tandems, Heidelberg, July 1966. 
[10] See for instance, H. A. Weidenmuller : Nucl. Phys., 85, 241 (1966); and A 99, 269, 

289 (1967); Proc. S.I.F., Course XL, Varenna (1967), p. 780; R. H. Lemmer: Proc. 

S.I.F., Course XL, Varenna (1967), p. 445; see also T. Tamura: Phys. Rev., 185, 

1256 (1969). 
[11] A. F. R. de Toledo Piza and A. K. Kerman: Ann. Phys., 43, 363 (1967); A. K. Ker- 

man and A. F. R. de Toledo Piza: Ann. Phys., 48, 173 (1968); see also A. M. Lane: 

Isospin in Nuclear Physics, Editor D. H. Wilkinson, North-Holland, Chapt. 11, 

p. 511 (1969). 



236 R. A. Ricci 

[12] D. Robson, J. D. Fox, P. Richard and F. C. Moore: Phys. Lett., 18,86(1965). 

[13] C. F. Moore and G. E. Terrell: Phys. Rev. Lett., 16, 804 (1966). 

[14] The first experiment of this type was done by C. F. Moore, L. J. Parish, P. von 

Brentano and S. A. A. Zaidi: Phys. Lett., 22, 616 (1966) following the original sug- 
gestion of G. A. Jones, A. M. Lane and G. C. Morrison: Phys. Lett., 11, 129 (1964). 

See also the recent work of P. von Brentano et ah: Phys. Lett., 26 B, 666 (1968) 

and of S. A. A. Zaidi et ah: Phys. Rev., 165, 1312 (1968). 
[15] P. Richard, C. F. Moore, D. Robson and J. D. Fox: Phys. Rev. Lett., 13, 343 (1964), 
[16] G. A. Keyworth, G. C. Kyker, E. G. Bilpuch and H. W. Newson: Nucl. Phys.. 

89, 590 (1966). 
[17] R. A. Ricci: Proc. Intern. Conf. Progress Nucl. Phys. with Tandem, Heidelberg, 

1966; G. Chilosi, R. A. Ricci and G. B. Vingiani: Phys. Rev. Lett., 20, 159 (1968); 

G. B. Vingiani, R. A. Ricci, R. Giacomich and G. Poiani: Nuovo Cimento, 57, 

453 (1968). 
[18] G. B. Vingiani, G. Chilosi and W. Bruyensteyn: Phys. Lett., 26 B, 285 (1968), 
[19] K. W. Jones, J. P. Schiffer, L. L. Lee, A. Marinov and J. L. Lerner: Phys. Rev.. 

145, 894 (1966). 
[20] P. Wilhjelm, G. A. Keyworth, J. C. Browne, W. P. Beres, M. Divadeenam, H. W. 

Newson and G. E. Bilpuch: Phys. Rev., 177, 1553 (1969). 
[21] G. A. Keyworth, G. C. Kyker, H. W. Newson and E. G. Bilpuch: Bull. Amer. 

Phys. Soc., 12, 585 (1967). 
[22] J. C. Browne, G. A. Keyworth, D. P. Lindstrom, J. D. Moses, H. W. Newson 

and E. G. Bilpuch: Phys. Lett., 28 B, 26 (1968). 
[23] C. Bloch and J. P. Schiffer: Phys. Lett., 12, 22 (1964). 
[24] E. Kashy, A. Sperduto, H. A. Enge and W. W. Buechner: Phys. Rev., 135 B, 865 

(1965). 
[25] W. Beres and M. Divadeenam: Nucl. Phys., 117, 143 (1968). 
[26] P. M. Endt: Nuclear Structure (North-Holland Publ., Amsterdam, 1967). 
[27] G. M. Temmer: Proc. Intern. Conf. Nucl. Struct., Dubna, 1968. 
[28] D. D. Watson, J. C. Manthuruthil and F. D. Lee: Phys. Rev., 164, 1399 (1967). 
[29] S. Maripuu: Nucl. Phys., A 123, 357 (1969). 
[30] I. Fodor et ah: Nucl. Phys., A 116, 167 (1968). 
[31] R. Giacomich, I. Lagonegro, R. A. Ricci and G. B. Vingiani: to be published. 



The Crab Nebula. 
Ancient History and Recent Discoveries. (*) 

B. B. Rossi 

Center for Space Research, M.I.T. - Cambridge, Mass. 



1. - The Chinese and Japanese chronicles for the year 1054 of the Chris- 
tian era registered the sudden appearance in the constellation of Taurus of 
a new star — a « guest star » — of extraordinary brightness, which gradually 
faded away until, some two years later, it was no longer visible. 

Centuries went by, and hardly anyone was aware of this event when, in 
1771, the French astronomer Messier compiled a catalogue of all known 
comet-like objects (nebulae and clusters) that appeared to occupy fixed 
positions in the sky. The first item on his list (Ml) was a nebula in the 
constellation of Taurus, about 4 arc minutes across, whose existence had 
been known for about 40 years. During the following decades this nebula 
was observed repeatedly with improved telescopes. In 1848 the shape of the 
object suggested to the British astronomer Lord Ross the name of Crab 
Nebula, which has been since generally accepted. 

The next event of crucial importance for the present story was the detection 
at a Baltic observatory, in 1885, of an exceedingly bright star in the Andro- 
meda galaxy, that was the result of a sudden flare up. In the subsequent 
years, a number of similar stellar outbursts were observed in external gal- 
axies. In some cases the brightness of the « new » star was comparable to 
or even greater than the total brightness of the galaxy before the outburst. 
By 1920, it had become generally accepted that these extraordinary out- 
bursts were not limiting cases of ordinary novae, but were to be regarded 
as an entirely different class of astronomical events. Since the late thirties 
these events have been known as supernovae. 

The discovery of supernovae in external galaxies stimulated the interest 
of astronomers in the historical records of events that might be interpreted 



(*) This work was supported in part by the National Aeronautics and Space Admin- 
istration under grant NGR 22-009-015. 



238 



B. B. Rossi 



as supernovae outbursts within our own galaxy, and prompted them to search 
for celestial objects that might be regarded as remnants of these outbursts. 
In the early twenties astronomers noticed the coincidence between the posi- 
tion of the Crab Nebula and the position of the « guest star » of 1054, as 
could be deduced from the descriptions contained in the oriental chronicles. 
They also discovered that the angular dimensions of the Crab Nebula were 
gradually increasing. Under the assumption that the nebula had originated 
from a point-like object and had undergone uniform expansion since its birth, 
it was possible to compute its age, which turned out to be close to the time 
elapsed since the appearance of the « guest star ». On the basis of these 
results Hubble, in 1928, suggested that this event had been a supernova out- 
burst, and that the Crab Nebula was its remnant. 

In the following years the very powerful optical telescopes which by then 
had become available were applied to a systematic study of the Crab. It was 




Fig. ]. - a) Picture of the Crab Nebula in « white light » (taken through a po- 
laroid filter), showing the diffuse luminosity. 



The Crab nebula. Ancient history and recent discoveries 



239 



found that this object consisted of an « amorphous mass », in which long 
and thin « filaments » were embedded. The light from the « amorphous 
mass» (which accounted for over 90% of the whole optical emission from 
the Crab) had a continuous featureless spectrum. In the light from ihc fila- 
ments, on the other hand, the lines of the known elements (especially the 
Hat line of hydrogen) appeared prominently (see Fig. I). 




Fig. I, - h) Picture of the Crab Nebula in Hx {taken through an interference 
filler).* showing Ihc filamentary structure (Mt. Wilson and Paloniar Observatories). 



The spectral lines of individual filaments were observed to exhibit Doppler 
shifts, which were interpreted as due to the expansion of the nebula. From 
this effect, the radial component of the velocity of expansion was found to 
be a little over 1000 km/s. This result together with the observed rate of 
increase of the angular radius (0.21 arc s/y) provided an estimate of 5000 



240 B. B. Rossi 

light years for the distance of the Crab Nebula, under the assumption that 
the velocities of expansion along the line of sight and perpendicularly to it 
were identical. (However, it is now believed that the expansion may not be 
exactly isotropic, and consequently that the above estimate of the distance 
may be in error by some 20%, probably on the low side.) 

In the meantime, theoretical ideas pertinent to the supernova phenom- 
enon began to emerge. Already in 1939 Oppenheimer and his collaborators 
addressed themselves to the problem of what happens when the nuclear fuel 
in the central part of a star is nearly exhausted, so that the pressure of the 
radiation generated by the nuclear reactions can no longer balance the forces 
of gravitational attraction. They found that, depending on its mass, the star 
will collapse either into a « white dwarf », or into a lump of nuclear matter, 
i.e., a « neutron star ». According to present views , prior to the final collapse, 
part of the stellar mass is blown out into space, perhaps because of the sudden 
ignition of the remaining nuclear fuel. This outburst manifests itself as a 
supernova, and the ejected matter forms the cloud later found at the location 
of the outburst. 

By then, astronomers had discovered two faint stars near the center of 
the Crab Nebula, and had suggested that either of them might be the residual 
condensed object of the supernova explosion of 1054. However, while one 
of these stars had an entirely « normal » spectrum, the other (known to the 
astronomers as the south preceding star) was found to have a featureless 
spectrum, quite different from the spectra of ordinary stars. Furthermore, 
for some time astronomers had been observing certain peculiar « ripples », 
which traveled through the cloud at enormous speed (about r the speed of 
light). Careful measurements showed that the vector velocities of these 
ripples were directed away from the south preceding star. For both these 
reasons Baade and Minkowsky in 1942 concluded that this object rather than 
the other member of the doublet should be identified as the supernova remnant. 

2. - Astronomical research in the years following the end of the second 
world war was dominated by the almost explosive development of radio 
astronomy. One of the first discrete radio sources to be identified with an 
optical object was the Crab Nebula [1]. 

The discovery of the radio emission of the Crab brought to a sharper 
focus the problem of the origin of the radiation from this object, which had 
puzzled astronomers for several years. Indeed, while it had been found very 
difficult to explain the shape and the intensity of the optical continuum in 
terms of thermal processes (the only celestial radiation processes well under- 
stood at the time), in no way could processes of this kind account for the 
strong radio signals. 



The Crab nebula. Ancient history and recent discoveries 241 

The solution of the problem came in the early fifties when Shklovsky 
suggested that both the radio emission and the optical continuum were due 
to the same, nonthermal process, a process to be identified with the so-called 
synchrotron effect, i.e., the emission of electromagnetic radiation by highly 
relativistic electrons traveling in a magnetic field [2]. Unlike thermal radiation, 
synchrotron radiation is linearly polarized. Although it was difficult to pre- 
dict whether or not a polarization might actually be observable (since in 
the case of a source of finite dimensions the net effect depends on the degree 
of randomness of the magnetic field), Shklovsky's suggestion prompted astron- 
omers to search for a polarization of the optical continuum of the Crab. 
The positive results of these observations, and the detection, some time later, 
of a similar polarization in the radio band of the spectrum, have been generally 
accepted as a crucial test of the synchrotron hypothesis. Today, of course, 
the synchrotron process is known to play a major role in many astrophysical 
phenomena. But it is worth noting that it was in the Crab Nebula that the 
occurrence of this process on a cosmic scale was first established. 

Synchrotron emission extending into the optical band implies that the 
Crab Nebula is permeated by a magnetic field (of an estimated strength 
between 10 -4 and 10~ 3 G) and contains electrons with energies extending 
up to at least 10 12 eV. Various suggestions about the origin of these electrons 
were put forward (although none was worked out quantitatively into a theory). 
High-energy electrons might have been left over from the original explosion; 
or they might be ejected continuously from the central star; or they might 
be accelerated while moving through the cloud by some sort of Fermi-type 
process. Whatever mechanism was responsible for the acceleration of elec- 
trons, it was thought that the same mechanism would also accelerate protons 
and heavier nuclei. While the electrons lost their energy (or most of it) within 
the cloud by synchrotron emission, protons and heavier nuclei (for which 
synchrotron losses are negligible) would escape into interstellar space without 
appreciable energy loss, and would thus contribute to the galactic cosmic-ray 
flux. In fact, it was argued that all galactic cosmic rays may originate from 
supernovae, being produced primarily at the time of the initial outburst. 

3. - In 1962, the discovery of surprisingly strong celestial sources of 
X-rays — including both localized sources and a diffuse background [3] — 
opened up the new field of X-ray astronomy. X-rays, of course, can only 
be observed at very high altitudes, because of their strong absorption in 
the atmosphere. Most of the results available to this date have been obtained 
by means of rockets, although balloons have made important contributions 
to the study of the « hard » component of the X-ray flux. The second X-ray 
rocket, flown in October 1962 [4] already gave some tentative indication 

16 



242 B. B. Rossi 

of an X-ray source in the general direction of the Crab Nebula. The follow- 
ing spring a rocket equipped with a detector of improved angular resolution 
established the existence of an X-ray source within a few degrees of the 
Crab [5]. The crucial proof that this source was indeed coincident with the 
Crab came in the summer of 1964 when a rocket flown during an eclipse 
of the Crab by the moon showed the simultaneous disappearance of the 
X-ray and of the optical flux [6], The identification was confirmed in 1967 
by means of a collimator of very fine angular resolution, which measured both 
angular coordinates of the X-ray source with a precision of about 20 arcs [7]. 
The results of the 1964 and 1967 observations are summarized in Fig. 2. 
They agree in showing that, within the observational uncertainties, the center 
of the X-ray source is coincident with the center or the visible nebula. More- 
over both experiments indicate that the X-ray source is not point-like, but 



NRL (MANLEY 1965) 



zrss'w- 




Fig. 2. - Observational results on the location and size of the X-ray source in 
the Crab Nebula, superimposed on a photograph of the nebula in ordinary 
light [7]. The data were obtained by Bowyer et at, [61 who observed the occult- 
ation of the Crab by the moon, and by Oda el at. [7], using a modulation col- 
limator. The arc marked «NRL 1964 » shows the position of the moon's limb 
at the time when it crossed the center of the X-ray source, as given by Bowyer 
et af. The arc marked «NRL (Manley 1965) » shows the same data, corrected 
for the motion of the rocket during the experiment [31]. The intersection of the 
« prerotl» and « postroll» lines is the most likely position of the center of the 
source, as determined by Oda et a!.\ the observational errors of this determi- 
nation are also indicated. The dotted circle represents the approximate dimen- 
sions of the X-ray source. 



The Crab nebula. Ancient history and recent discoveries 243 

has an angular diameter of about 100 arcs {i.e., of the same order as that 
of the visible nebula, although perhaps somewhat smaller). 

Since its discovery, the X-ray source in the Crab has been the object of 
many observations. In reporting the results of these observations, it may be 
instructive to compare them with those concerning another strong X-ray 
source, ScoX-1, which has also been extensively investigated. Unlike the 
Crab, Sco X-l had not been recognized by the astronomers as a peculiar 
celestial object before its discovery as an X-ray emitter. Subsequently it was 
identified with a faint star of unusual spectral characteristics [8]. Again 
unlike the Crab, Sco X-l appears point-like (to the limit of the resolution 
achieved so far) both in the optical and in the X-ray band. 

The X-ray emission from the Crab, as well as its light emission, were 
found to be nearly constant in time, at least when averaged over periods of 
seconds (Sco X-l, on the contrary, was found to be highly variable both in 
the X-ray and in the optical bands). 

In the X-ray band, the spectral function of the Crab (energy flux per unit 
interval of photon energy) was found to follow closely a power law with expo- 
nent close to unity from hv=l keV to A^=100keV. (The X-ray spectrum 
of Sco X-l has a very different shape, being represented approximately by 
an exponential function, similar to that expected from a thermal, optically 
thin source at about 5xl0 7 °K. This implies that the spectrum of Sco X-l 
is much « softer » than that of the Crab; indeed, while Sco X-l is about 
10 times brighter than the Crab at photon energies of the order of 5 keV, 
the Crab becomes brighter than Sco X-l at photon energies above about 
30 keV.) 

A log-log plot of measurements in the radio, visible, ultraviolet, and X-ray 
bands suggests that the whole electromagnetic spectrum of the Crab may 
be described by a single smooth function. This has been taken as an argu- 
ment in favor of a common origin (i.e., synchrotron radiation) for the entire 
spectrum. Although not yet definitely proven, the assumption of a synchro- 
tron origin for X-ray spectrum of the Crab is accepted by most scientists, 
to a large extent because of the difficulty of finding a more likely alternative. 
The only other process that has been considered seriously is thermal radiation 
from a hot, optically thin plasma cloud. As already noted, if the cloud is 
at a uniform temperature, this process gives rise to an exponential spectrum, 
i.e., a spectrum more similar to that of Sco X-l than to that of the Crab. 
Of course, if the plasma temperature varies from point to point, as it may 
well do in the Crab Nebula, the X-ray spectrum will be a sum of exponentials 
which might conceivably simulate a power law over a limited range of photon 
energies. However, beyond a photon energy corresponding to the temper- 
ature of the hottest region, the spectrum should drop sharply. Therefore 



244 B. B. Rossi 

the possibility of a thermal radiation process became increasingly remote as 
spectral measurements were extended to higher and higher energies and 
failed to detect any cut-off. 

With the magnetic fields that supposedly exist in the Crab, synchrotron 
emission in the X-ray band requires electron energies of the order of 10 14 eV. 
It is worth noting that for these very energetic electrons the synchrotron 
process is exceedingly effective. Consequently the electrons lose energy at 
a very fast rate, which appears to rule out the possibility that they might 
have originated from the initial explosion. 

At this point it may be useful to quote some figures. The X-ray flux 
from the Crab Nebula, in the spectral band from hv = 1 keV to hv=l00 keV, 
amounts to about 7 x 10~ 8 erg/cm 2 s at the earth. Taking the distance 
of the Crab as 5000 l.y., its X-ray emission turns out to be about 2xl0 3r 
erg/s, i.e., about 5000 times the total emission of the Sun in all wavelengths. 
The emission in the optical band is about \ and the emission in the radio 
band (A>3cm) is about 1/1000 of the X-ray emission. (For ScoX-1, the 
corresponding figures are about 1/1000 and about 2xl0 -8 .) 

4. - We now come to the very recent developments of astronomical 
research, and here again we find that the Crab Nebula occupies a central 
position in the new discoveries. 

Early in 1968, Hewish and his co-workers announced the discovery of 
pulsating radio sources, or pulsars [9]. At the end of that year, some 25 
pulsars were known, with periods ranging from about 2 to 1/30 s. Of these, 
only two had been identified with previously known celestial objects, both 
of them supernova remnants. One of them was Vela X [10] the other was 
the Crab Nebula [11]. The pulsar in Vela X had a period of about 89 ms, 
that in the Crab (known also as NP 0532) had a period of about 33 ms, 
the shortest among all known pulsars. 

The periods of the « slow » pulsars were found to be remarkably constant 
(for some of them it was established that the rate of change was less than 
one part in 10 8 per year). The periods of the « fast » pulsars in Vela X and 
the Crab, on the other hand, were found to increase very slowly. For the 
Crab, the rate of increase amounts to one part in 2400 per year (*). 

In January 1969 another important discovery took place, with the detection, 
in the Crab Nebula, of the first and thus far the only optical pulsar [14]. 
The period of the optical pulsations was found to be exactly identical to that 
of the radio pulsations, which proved beyond any reasonable doubt that the 



(*) In the case of Vela X, the gradual increase of the period was interrupted, between 
February 4 andh March 3, 1969, by a sudden decrease of two parts in one million [12, 13]. 



The Crab nebula. Ancient history and recent discoveries 



245 




Fig. 3. - Siroboscopic pictures of the stars near the center of the Crab Neb- 
ula taken by J. S- Miller and E. I. Wamplcr at the Lick Observatory. The 
pulsar appears as the brightest object in the picture at the top; it is nearly in- 
visible in the picture at the bottom. The change in the apparent brightness is 
due to the gradual phase change of the light pulses relative to the « open pe- 
riods » of the stroboscope disk [16] (Lick Observatory photograph). 



246 B. B. Rossi 

radio and the optical pulsars were the same object (although, of course, the 
radiations belonging to the two spectral bands may come from different 
regions of this object). Precise determinations of its position showed that 
the pulsating star is the south preceding member of the doublet found near 
the center of the Crab [15], and thus confirmed unequivocally the previous 
tentative identification of this star as the condensed residue of the supernova 
explosion. A further dramatic verification of this result came from a series 
of photographs taken through the slots of a rotating disk, which showed 
that the brightness of the south preceding star changed periodically between 
a maximum and practically total extinction when the time between successive 
« open » intervals was nearly equal to the period of the pulsations (see Fig. 3). 

Quite naturally, the discovery of the optical pulsar in the Crab suggested 
a search for a pulsating component in the X-ray emission of the same object. 
During the month of April 1969, two rockets provided with detectors sensitive 
to « soft » X-rays (photon energies of several keV) were launched for this pur- 
pose, the first by the NRL group [17], the second by the MIT group [18]. 
Both experiments did, in fact, detect the expected pulsations, with a 
period exactly equal to that of the radio and of the optical pulsations 
(33.099522 ms at the time of the MIT flight). 

Finally, a recent analysis of balloon data obtained in 1967 revealed that 
also the « hard » X-ray flux of the Crab (photon energies greater than about 
35 keV) contains a pulsating component [19]. A balloon flight carried out 
in May 1969 confirmed this result and provided quantitative information 
on the size and shape of the pulses [20]. 

Examples of the pulse shapes observed in different spectral bands appear 
in Figs. 4-7. Shown in each case is the time dependence of the radiation flux 
during one period, averaged over a large number of periods. 

One sees that, at all wavelengths, each pulse contains two peaks, sepa- 
rated by a time interval slightly less than one half the period. In the optical 
and in the X-ray bands, the shape of the pulses appears to be quite constant. 
In the radio band, however, the pulse shape varies greatly from pulse to 
pulse, and even averaging over thousands of pulses does not result in a stable 
pattern. It has been pointed out that this instability may be due, at least 
in part, to refraction of radio waves, possibly in the ionized gases within the 
nebula itself [23, 24]. This interpretation is consistent with the observed sta- 
bility of the optical and of the X-ray pulses because refraction effects decrease 
rapidly with decreasing wavelengths. 

There is evidence that at all wavelengths the radiation level between the 
first and the second peak is somewhat higher than the radiation level after 
the second peak. We shall take the view that this lowest level of radiation 
represents the steady emission of the nebula. In other words we shall assume 



The Crab nebula. Ancient history and recent discoveries 



247 



that the emission of the pulsar actually drops to zero during each period. 
(Stroboscopic pictures such as those shown in Fig. 3 tend to support this 
assumption, but do not prove that it is rigorously correct.) By taking the 
lowest radiation level as the zero line, we can then separate the pulsating 
component of the radiation originating from the pulsar, from the steady 
component originating from the nebula. 




10 ms 





10ms 

Fig. 4. - Average pulse shapes of the pulsar in the Crab Nebula, as observed 
on three different days and at three different radio frequencies with the 1000 ft 
antenna at the Arecibo Ionospheric Observatory; a) Nov. 14, 1968; 196.5 MHz; 
18000 pulses, b) Nov. 26, 1968; 198 MHz; 21 153 pulses, c) Dec. 2, 1968; 
430.0 MHz; 53 427 pulses [21]. 



248 



B. B. Rossi 



7,000 






•"■ " ' t 


1 1— 




— i 1 


1 r 


6,000 


■ 












■ 


5,000 














- 


4,000 










a 




- 


3,000 




." 


"-.__ 


/ 






■ 


2,000 






'' : -r>.».^,. 


-•.•..-v.v/.-w.-----''" 




^,,0^,;^,,.. 


,,,v-,,... 






.-' 






b 




- 








"'•"---..,.._....._.... . 








- 


250 










a-b 




- 







' 


-i '. L 1_ 


1 1 


, •'••."■•'■:" 







75 100 125 150 175 200 225 250 275 



Fig. 5. - Light curves for the Crab pulsar in white light, a) sum of 100000 pe- 
riods; b) sum of 30000 periods, taken 3 J h earlier. The abscissa is channel 
number, each channel being of 100 \xs duration; the left-hand scale refers 
to curve a) and the right-hand scale to curve b) [22]. 



Observations show that the ratio between the power in the pulsating 
mode and the power in the steady mode varies by a very large factor over 
the spectrum. In the radio and in the optical bands this ratio amounts to 
only several parts in one thousand. In the « soft » X-ray band it reaches 
the value of about 9 % and in the « hard » X-ray band it seems to be higher 
still. From these results and from the spectral data on the total emission 
of the Crab reported previously it follows that all but a minute fraction 
(perhaps less than 1 %) of the radiation from the pulsar is in the form of 
X-ray. This object, then, may be properly described as an X-ray pulsar. 

The pulses observed in the optical and the X-ray bands, while very dif- 
ferent in their size relative to the steady component, have strikingly similar 
shapes. In both spectral bands, one of the two peaks observed during each 
period has a width of about 1.5 ms, and the other has a width of about 
3.5 ms (*). Within the experimental errors, the separation of the two peaks 



(*) However, one should note that the observed width of the narrow X-ray peak is 
not much greater than the time resolution of the instrument. 



The Crab nebula. Ancient history and recent discoveries 



249 



(0) X-RAYS 
8000 h- fil 



o 
o 

a. 

0) 



7000 - 



6000 - 




4.5% » 



13 3 msec f 



A 



4>6% 



7490 
counts 



IT 

Primary 

Optical 

Peak 



I" 




4i 



B 



counts _ ^J 



_[b) Optical 




33,099,522 nsec 



Fig. 6. - a) «Soft» X-ray data for the Crab pulsar obtained during 150 s 
of the rocket flight carried out by Bradt and his co-workers on April 27, 1969. 
The detector was sensitive to photons in the energy range from 1.5 to 10 keV. 
Data were superimposed by dividing each period into 40 equal « bins » and 
distributing the counts into these bins. The intensities represented by the areas 
under peaks A and B are 4.5% and 4.6%, respectively, of the total X-ray in- 
tensity from the Crab, b) Optical data shown in Fig. 5, integrated into 41 « bins » 
for comparison with the X-ray data [18]. 



is the same (about 13.5 ms). In the experiment by Bradt and his co-workers 
(see Fig. 6) recording of time signals from the WWV radio station during 
the rocket flight made it possible to correlate the X-ray observations with 
optical observations carried out, within a few hours of the flight, at the Mc 
Donald Observatory and at the Palomar Observatory. It was thus shown 
that the narrow peaks in the X-ray and in the optical bands are simultaneous 
within 1 ms. 



250 



B. B. Rossi 



20 200 - 



c 20 000 h 
m 

w. 

°- 19 800 h 



o 
u 



19 600 - 



19 400 



I4±|ms 




2941 
counts 



I— ------ u,- 




I I I I I I t I I I I I I I I I I I I I I I I I I 



I 3 5 7 9 II 13 15 17 19 21 23 25 27 29 
Bin Number 

Fig. 7. - « Hard » X-ray data for the Crab pulsar obtained during the balloon 
flight of May 10, 1969 by Floyd and his co-workers. The measurements cover 
the energy range from 25 to 100 keV. The data are divided into 30 « bins » [20]. 



The great variability of the radio pulses denies the possibility of a detailed 
comparison of their shape with that of the optical and radio pulses. Further- 
more, the wavelength-dependent delay of the radio pulses due to dispersion 
in the interstellar medium makes it difficult to establish an exact time cor- 
relation between the radio peaks and the optical peaks. All one can say 
on the basis of published reports is that the peaks in the radio and optical 
bands are simultaneous, with an uncertainty of about 6 ms, due almost 
entirely to the interstellar dispersion [25]. 



5. - A reliable theoretical interpretation of the observational data that 
have been described above is still lacking. From these data, however, there 
begins to emerge a model which, although tentative and incomplete, may be 
worth discussing. 

When pulsars were first discovered, two different kinds of models were 
suggested to account for their equally-spaced signals; i.e., a) vibrational 
models and b) rotational models. The vibrating or rotating star was thought 
to be either a) a white dwarf or b) a neutron star. While it was difficult to 
discriminate between these various possibilities as long as only pulsars with 
periods of the order of a second were known, the discovery of pulsars with 
periods of less than 0.1 s practically eliminated all choices but one. Since 
the free oscillations of white dwarfs have periods considerably longer than 



The Crab nebula. Ancient history and recent discoveries 251 

0.1s; since white dwarfs cannot rotate at 10 rps or more without being 
disrupted; since the free oscillations of a neutron star are believed to be 
rapidly damped through the production of gravitational waves, it became 
practically certain that pulsars (or at least the « fast » pulsars such as that 
in the Crab Nebula) were rotating neutron stars. 

We can estimate the kinetic energy of rotation E of the pulsar in the 
Crab by assuming that its mass is of the order of one solar mass (~ 2 x 10 33 g) 
and by taking the conventional value of 10 km for its radius. With the 
observed angular velocity of 27zx30^ 190 s _1 we obtain 

E^ 1.4xl0 49 erg. 

From this figure and from the observed rate of increase of the period it 
follows that the pulsar loses rotational energy at the rate 

dF 

— ~ t*t 3.7 xlO 38 ergs^ 1 . 
at 

From the data reported previously we may estimate the total energy of the 
electromagnetic radiation of all frequencies emitted by the Crab Nebula to 
be several times 10 37 erg s _1 . It seems likely that an amount of energy, 
perhaps of the same order of magnitude, may be spent by the Crab Nebula 
in the production of cosmic rays. Thus, within the large uncertainties of 
the present estimates, — dE/dt appears to be remarkably close to the total 
energy output of the Crab Nebula, which naturally suggests that this energy 
is supplied the gradual slowing down of the rotating neutron star at the center 
of the Crab [26, 27]. An additional justification for accepting this suggestion 
as a working hypothesis in the formulation of our model may be found in 
the fact that previously it had been necessary to resort to ad hoc assumptions 
in order to account for the energy storage in the Crab Nebula. 

It appears natural to interpret the pulsating signals received from a rotat- 
ing object as due to a light-house effect [28]. As another working hypothesis, 
we shall therefore assume that the electromagnetic radiation from a neutron 
star is confined to one or more narrow beams, which sweep past the observer 
as they corotate with the star. In the case of the Crab, there would be at least 
two such beams. The narrow principal peak requires a beam whose angular 
width in the direction perpendicular to axis of rotation is at most 2tz/20 (less 
if the axis of rotation is not perpendicular to the line of sight.) If this beam 
were in the shape of a circular cone, the a priori probability of its being de- 
tected by an observer on the earth would be 5% or less. Similarly the prob- 
ability of detecting the beam responsible for the wider pulse would be 10 % 



252 B. B. Rossi 

or less. We conclude that either the earth is in a peculiarly favourable po- 
sition for the observation of the signals from the pulsar in the Crab ; or the 
beams responsible for these pulses are fan-shaped rather than circular; or 
there are more than two beams. 

The emission of the radiation into discrete beams implies an azimuthal 
anisotropy in the structure of the pulsar with respect to its spin axis. The 
stability of the beams as observed in the visible and X-ray bands is more 
easily understandable if the anisotropy is due to a magnetization of the pulsar 
rather than to « hot spots » or other peculiarities in a plasma atmosphere 
of the pulsar, as had been suggested when only radio observations were 
available [28]. It should be noted that the collapse of a star with a moderate 
magnetic field will, indeed, result in a neutron star with exceedingly large 
magnetization, even if only a minor fraction of the original magnetic flux 
is conserved. (For a star similar to the Sun, 100% flux conservation would 
give rise to fields of the order of 10 9 G at the surface of the neutron star; 
field strengths up to 10 13 G have been mentioned as a possibility.) It thus 
appears reasonable to further specify our model by assuming that the neutron 
star is strongly magnetized, and that the magnetization is not axially symmetric 
with respect to the spin axis. 

We now come to the problem of the processes responsible for the steady 
component of the radiation (originating from the nebula) and of the pulsat- 
ing component (originating from the neutron star). With regard to the former, 
as already noted, we know for sure that the continuous spectrum extending 
from the radio waves to the ultraviolet is due to a synchrotron effect, and we 
have good reasons to believe that the same effect is also responsible for the 
X-ray emission ; which means that the nebula contains electrons with energies 
up to at least 10 14 eV. According to our model, these electrons derive their 
energy from the kinetic energy of rotation of the neutron star. We may think 
of a direct process, whereby the electrons are accelerated by the strong time- 
varying electromagnetic field that exists in immediate neighborhood of the 
star, and are then injected into the surrounding magnetized plasma cloud. 
Alternately, we may think of an indirect acceleration mechanism; i.e., we may 
assume that the rotating neutron star loses energy to the cloud giving rise to 
disturbances (in the form of waves or shocks), which then, through a Fermi- 
type stochastic interaction with the electrons in the cloud, supply the energy 
radiated via the synchrotron process. 

An analysis of the stochastic acceleration process (for example on the 
basis of a model based on the interaction between Alfven waves and individual 
electrons [29]) shows that the high efficiency needed to maintain the required 
electron spectrum can be achieved only under rather extreme circumstances. 
On the other hand, no quantitative treatment of the direct acceleration 



The Crab nebula. Ancient history and recent discoveries 253 

process has yet been developed. In this connection one should keep in mind 
that the electrons will lose energy by synchrotron radiation even as they 
are accelerated ; and that the synchrotron losses are proportional to the square 
of the magnetic field and to the square of the energy. Therefore it is not 
easy to figure out how electrons can emerge from the region of strong ma- 
gnetic field surrounding the neutron star with the enormous energy they need 
to radiate X-ray photons in the weak field of the nebula. 

Let us consider next the pulsating component of the radiation. One may 
think of a variety of processes capable of generating pulsations in the long- 
wavelength band of the spectrum. The fundamental problem, however, is 
to explain the emission in the X-ray band which, by itself, accounts for at 
least 99 % of the pulsating power, as already noted. In this portion of the 
spectrum, it appears that the only effective emission process is the inter- 
action of electrons with the magnetic field. This process presupposes the 
existence around the neutron star of electrons with a suitable energy distri- 
bution. In the frame of reference corotating with the star, the spacial 
distribution of the electrons must be remarkably stable; i.e., the electron 
cloud must corotate rigidly with the star. Furthermore, the distribution of 
the electrons in velocity space, and the pattern of magnetic field lines, must 
be such as to account for the required beam-shaped emission. 

Of course, rigid corotation can only occur up to a maximum distance of 
the spin axis where the rotational velocity becomes equal to the velocity of 
light [28]. With an angular velocity of 190 s~\ this distance amounts to 
1.6xl0 8 cm. Note that, if the magnetic field resembles that of a dipole, 
and therefore varies as the inverse cube of the distance, its magnitude at 
the « light circle » in the equatorial plane is about 2.5 x 10 4 times smaller 
than at the surface of the neutron star. 

Of course, electrons require a much smaller energy to radiate X-ray 
photons in the strong magnetic field surrounding the neutron star than they 
do in the weak magnetic field of the cloud. In this connection it is important 
to keep in mind that the motion of electrons in the plane perpendicular to 
the magnetic field is actually quantized [30]. In the subrelativistic region 
the energy levels are equidistant with a separation Ae = hco/2ji, where co 
is the cyclotron frequency. With Ae measured in eV and the magnetic 
field B in gauss, the following relation holds 

Ae = 1. 16xl0- 8 B. 

If Ae is very small compared with the photon energy, quantum effects are 
negligible and the interaction of the electrons with the magnetic field may 
be described by the classical theory of magnetic bremsstrahlung. In this 



254 B. B. Rossi 

case the average energy of the radiated photons is much smaller than the 
electron energy. If, however, Ae is close to the photon energy then the 
emission occurs via a process similar to an atomic quantum transition between 
two bound levels, and the energy of the emitted photons is equal to or a 
sizeable fraction of the electron energy. Even hard X-ray photons, then, may 
be produced by subrelativistic electrons. 

Quantized emission in the X-ray band requires magnetic fields of the 
order of 10 12 G or more. While these fields are not ruled out, it appears 
more likely to the author that X-rays are produced in a region of lower 
magnetic field, in which case relativistic electrons are needed. One must then 
assume that electrons are first accelerated to relativistic, but not necessarily 
extremely high, energies by the rotating neutron star. While in the vicinity 
of the star, they partake of its rotation and generate the pulsating component 
of the radiation. They then diffuse into the surrounding cloud, where, after 
perhaps gaining further energy, they give rise to the steady radiation. 

6. - To summarize, the model developed here pictures the Crab 
Nebula as a thin plasma cloud containing a weak magnetic field, with a fast- 
rotating, strongly-magnetized, neutron star at its center. The magnetization 
of the star does not have axial symmetry with respect to the spin axis, so 
that the rotation gives rise to time-varying electromagnetic fields, which, 
in some way or another, are capable of accelerating electrons. For a while 
these electrons remain within the corotating magnetosphere of the neutron 
star, where they give rise to corotating beams of electromagnetic radiation. 
Subsequently they diffuse into the surrounding cloud, where perhaps they 
acquire further energy by a Fermi-type stochastic process. Synchrotron 
emission by these electrons in the weak magnetic field of the cloud gives 
rise to the steady flux of radiation. 

Presumably the kinetic energy of rotation of the neutron star was initially 
derived from the conversion of some fraction of the gravitational energy 
released during the stellar collapse following the supernova explosion. From 
the time of its birth, the Crab Nebula has drawn from the rotating neutron 
star the energy needed to produce the various kinds of rays which it has been 
pouring out into space. 

Whether or not the general features of this model will survive future 
observations and future theoretical discussions is still an open question. 
Here the model is presented as a working hypothesis, that may be useful in 
suggesting further lines of investigation. From the theoretical point of view, 
one of the basic problems is clearly a quantitative analysis of the possible 
mechanisms for the acceleration of the electrons. From the observational 
point of view, it would be desirable to examine the polarization of the X-ray 



The Crab nebula. Ancient history and recent discoveries 255 

emission in order to test the assumption that it originates from a synchrotron 
process. Furthermore it would be very illuminating to extend the obser- 
vations of the steady and of the pulsating components of the electromagnetic 
spectrum to considerably higher photon energies. Finally we may hope that 
high-resolution X-ray pictures of the Crab, possibly taken at different wave- 
lengths, will furnish important information on the mechanism responsible 
for the acceleration of electrons and help discover the region of space where 
this acceleration occurs. 



REFERENCES 



[1] J. Bolton and G. Stanley: Aust. J. Sci. Res., A 2, 139 (1949). 
[2] I. S. Shklovsky: A. Zh., 30, 15 (1953); D.A.N. , 90, 983 (1953). 
[3] R. Giacconi, H. Gursky, F. R. Paolini and B. Rossi: Phys. Rev. Lett., 9, 439 (1962). 
[4] H. Gursky, R. Giacconi, F. R. Paolini and B. Rossi: Phys. Rev. Lett., 11, 530 (1963). 
[5] S. Bowyer, E. T. Byram, T. A. Chubb and H. Friedman: Nature, 201, 1307 (1964). 
[6] S. Bowyer, E. T. Byram, T. A. Chubb and H. Friedman: Science, 146, 912 (1964). 
[7] M. Oda, H. Bradt, G. Garmire, G. Spada, B. V. Sreekantan, H. Gursky, R. Giac- 
coni, P. Gorenstein and J. Waters: Ap. J., 148, L5 (1967). 
[8] A. R. Sandage, P. Osmer, R. Giacconi, P. Gorenstein, H. Gursky, J. Waters, 
H. Bradt, G. Garmire, B. V. Sreekantan, M. Oda, K. Osawa and J. Jugaku: 
Ap. J., 146, 316 (1966). 
[9] A. Hewish, S. J. Bell, J. D. H. Pilkington, P. F. Scott and R. A. Collins: Nature, 
217, 709 (1968). 
[10] M. I. Large, A. E. Vaughan and B. V. Mills: Nature, 220, 340 (1968). 
[11] D. H. Staelin and E. C. Reifenstein: Science, 162, 1481 (1968). 
[12] V. Radhakrishnan and R. N. Manchester: Nature, 222, 228 (1969). 
[13] P. E. Reichley and G. S. Downs: Nature, 222, 229 (1969). 
[14] W. J. Cooke, M. J. Disney and D. J. Taylor: Nature, 221, 525 (1969). 
[15] R. Lynds, S. P. Maran and D. E. Trumbo: Ap. J., 155, L 121 (1969). 
[16] J. S. Miller and E. J. Wampler: Nature, 221, 1037 (1969). 
[17] G. Fritz, R. C. Henry, J. F. Meekins, T. A. Chubb and H. Friedman: Science, 

164, 709 (1969). 
[18] H. Bradt, S. Rappaport, W. Mayer, R. E. Nather, B. Warner, M. Mac Farlane 

and J. Kristian: Nature, 222, 728 (1969). 
[19] A. J. Fishman, F. R. Harnden and R. C. Haymes: Ap. J., 156, L 107 (1969). 
[20] F. W. Floyd, I. S. Glass and H. W. Schnopper: Nature, 224, 50 (1969). 
[21] J. M. Comella, V. D. Craft, R. V. E. Lovelace, J. M. Sutton and G. L. Tyler: 

Nature, 111, 453 (1969). 
[22] B. Warner, R. E. Nather and M. Mac Farlane: Nature, 222, 223 (1969). 
[23] P. A. G. Sheuer: Nature, 218, 920 (1968). 
[24] O. B. Slee, M. M. Komesaroff and P. M. McCullough: Nature, 219, 342 (1968). 



256 B. B. Rossi 

[25] E. K. Conklin, H. T. Howard, J. S. Miller and E. J. Wampler: Nature, 111, 

552 (1969). 

[26] T. Gold: Nature, 111, 25 (1969). 

[27] A. Finzi and R. A. Wolf: Ap. J., 155, L 107 (1969). 

[28] T. Gold: Nature, 218, 731 (1968). 

[29] O. Manley and S. Olbert: Ap. J., 157, 223 (1969). 

[30] H. Y. Chiu, V. Canuto and L. Fassio-Canuto : Nature, 111, 529 (1969). 

[31] O. Manley, R. Finn and G. Ouellette: Private communication (1965). 



The K L — K s Mass Difference. 



C. RUBBIA 
CERN - Geneve 



1. - Introduction. 

The existence of a very small mass difference between the long- and the 
short-lived components of the neutral K system has been firstly postulated 
in the theory of particle mixture of Gell-Mann and Pais [1]. 

The K and K particles, produced in strong and electromagnetic inter- 
actions, belong to the isotopic spin doublets (K + , K ) and (K~, Kq) and have 
definite hypercharges, +1 and — 1, respectively. Furthermore, as a conse- 
quence of CPT invariance, particle and antiparticle have identical masses, 
i.e., M Ko -Mk o . 

However, weak interactions do not conserve hypercharge and they can 
induce not only decays but also transition between K and Kq. For instance 
transitions are mediated to the second order by any common decay chan- 
nel F, i.e. 

K -<-> F<-> K . 

After an infinitesimal proper time interval Sf, the hypercharge eigen- 
states K and K have evolved as follows [2]: 



(1) 



\Ky->\Ky-m{M\Ky+ b\k>}, 

\K> -> \K> — i$t{ A\K> + M\K)} . 



The (complex) terms A and B, represent the K <-> K mixing effects and 
M and M are related to the K and Kq masses decay rates: 

17 



258 C. Rubbia 

There exist two definite linear combinations of the \K} and liO states: 



(2) 



\L> = p\K> + q\K> , 
\Sy = r\K> + s\K> , 



which have simple, uncoupled time evolutions: 

\l> -> \Ly — i8tM L \L> , \sy -> \Sy—i8tM s \S} 

and which obviously describe the long- and short-lived states of simple 
exponential decay observed experimentally. The quantities M L and M s are 
defined : 

M L = m L — l -r L , M s = m s —-r s , 

where r L , m L , r s , m s are, respectively, the masses and decay rates of the 
long- and short-lived states. Relating expressions (1) and (3) with the help 
of formula (2) one finds easily: 

M = (M L sp — M s qr)/(sp —qr), M= (M s sp — M L qr)/(sp — qr) , 
A =(M s — M L )7tp/(sp — qr), B = (M L —M s )sq/(sp—qr) . 

Introducing the condition M = M required by CPT invariance and with 
some phase conventions, the above expressions become considerably simple, 
since s = — q, r =p: 

M=M = ^(M S +M L ), 

A = \^{M L -M S ), B=^(M L -M S ). 

The masses of the long- and short-lived states are therefore different, 
their difference being proportional to the terms A and B. As we have seen, 
these terms are originated by K <->K transitions which now give different 
contributions to the K L and K s self-energy diagrams. 

A first estimate of the magnitude of Bm = m L — m s is easily obtained 
from dimensional considerations [3]. If weak interactions to the first order 
satisfy to the rule |A»S| = 1, it is possible to connect K and K only with 
second or higher order diagrams. Consequently Sm is of order (Grsinfl) 2 , 
where Gf is the Fermi constant, and 6 is the Cabibbo angle: 

G 2 

8w c± --^ sin 2 • m 5 ~ 10~ 5 eV (for m = w K ) . 

4?r 



The K L — K s mass difference 259 

This is of the same order of magnitude of the total K s transition rate, 
r s = 1.145 x 1(T s-\ equivalent to T s h = 0.58 x lO" 5 eV. 

If K <-»K transitions occur directly (AS = 2) with strength f-G?, the 
corresponding mass difference is approximately: 

8ra =/G F m 3 ~ 1.2 x 10 3 /eV (for m = m K ) . 

Initially the interest in the K L — Kg mass difference has been simply 
whether it was of the same magnitude as r s h or much larger. Since the 
discovery CP violation in neutral K-decay [4] it has become important to 
know with precision the mass difference. 

An accurate knowledge of Sra is at the basis of the phenomenological 
analysis of the nature of CP violation in neutral decay. A more accurate 
value of Sra is also demanded in the analysis of experiments measuring the 
phase of the CP violating amplitude r} + _ = A(K l -+-k + k-)/A(K s ->tz + iz-). 
The most direct way to determine the phase of rj ± is to observe interference 
between the K l ^-k + -k~ and K s ->tz + tz~ decay amplitudes close to the pro- 
duction point of K state [5]. Since \rj + _\ ~ 2x 10~ 3 , such interference effects 
are large only relatively far from the production point where the K s amplitude 
has decayed down to the |»? + _| level: exp|T g f/2) ~r) + _. 

The error in the measured phase then will come to the greatest extent 
from the error in Sra which enters in the argument of the interference term 
in the form <$m~t, with t~ 12 /r s . This technique is then limited to the accu- 
racy with which the mass difference is known. 



2. - How to measure the mass difference? 

Many different tecniques have been employed to determine 8ra, since 
the existence of such a mass difference was suggested. All these experiments 
show the mass difference as an interference process between a coherent super- 
position of K L and K s states, the phase difference between the two states 
showing a characteristic precession due to the mass difference Sra. The 
attractive feature of all these experiments is that the smallness of the K £ — K s 
mass difference makes it possible to realize experimental conditions in which 
a coherent superposition of K L and K s states can be observed over long times 
and over a variety of arrangements. In the present paper we shall consider 
only the most recent techniques which almost invariably make use of the 
phenomenon of coherent regeneration. Pais and Piccioni [5] first pointed 
out the existence of a regeneration process for neutral K mesons but it was 
M. L. Good [6] who gave a complete description of this process and the 
application of this technique to the determination of Sm. Whenever a beam 



260 C. Rubbia 

of long-lived K meson traverses a slab of material, the « regenerator » because 
of difference of strong interaction cross-section for K and K components, 
the propagation of the incoming state is modified and a small coherent K s 
amplitude is regenerated. Thus the time evolution of the K and K states 
instead of (PCT) 

\Ky^\K} — iM$t\Ky — iB$t\Ky, 

\K> -> \K> — iM8t\K> — iA$t\K} , 
we shall have in matter the additional terms due to nuclear absorptions 



where N is the number of nucleus for unit volume and a and a are the total 
cross-sections for K and K . In terms of the forward scattering amplitudes 
/ and /, from the optical theorem, a = (4n/k) Im (/) and a = (47i/k)Im(f). 
If the forward amplitude is not purely imaginary, the beam is not only atte- 
nuated but also refracted, then instead of /Im(/) and ilm(f) we should 
write simply / and /. Then 

| JO -> |/0 - 1& W> - 2 ^-fY t l*> + B \v) > 

\K> -> |JO - iBt {m|x> - ^fY t \K> + A \ K >} • 

Taking p times the first equation ± q times the second we obtain 

\Ly^\L>-m{M' L \Ly~(f-f)Nr s A s \s>}, 

\S»^\S>-i*t{Ml\S?>-%(f-f)Nr a A B \L>Y 



where 



M' L =M L -y(f+f)Nr s A s , 



M' s =M' s -^{f+f)Nr s A s , 

and $zf$t = r s A s = k/mjt, where r s is the K s decay length. 

Therefore, if f¥=f, \L} and \S) do not propagate independently but they 
are coupled. To the first order in a finite slab of material after a time r, 



The K L — K s mass difference 261 

the state becomes: 
|L> ^exp [iM' L x\\V>~ J^I M Nr s A s (pxp [-/M^J-exp [-iM' L r])\Sy . 

Thus there is a coherent regeneration of the state |»S>. The state at the 
exit of the plate is of the general form: 

\L> + q\S>, 
where q, the regeneration amplitude, is proportional to 

% M f L Z f M s Nr sMl-™P [-(M s -M L )r]) . 



3. - Single regenerator experiments. The K^-K^ interference in the n + n~ 
decay channel. 

It was realized promptly after the discovery of CP violation that the exi- 
stence of both K L ->^ + 7r _ and K s ^tz + tz- decays provides a very powerful 
method for investigating the K^— K s mass difference. Let us consider the 
forward going beam after a slab of material. As we have just seen the state 
after the proper time interval t, measured from the exit face of the regen- 
erator is: 

|out, O = exp [— iM L t]\L} + q exp [— iM s t]\S) . 

Since both |L> and \S} states decay into the state ti+tt with amplitudes 
<7r+7r _ |T|L> and <7r+7r-|7"|5'>, the decay amplitude for the state |out, t} is: 

<jr f jr-|r|out, t} = exp [— iM L t](ji+n-\T\V) + q exp ]— iM s tK7i+7i-\T\S} 

and the decay rate per unit of time is promptly obtained by squaring the 
above expression: 

dN 

w (0 = r(K s ->7T+o{|^ex P [-r s t] + 1*? + _|2 exp [-r L t] + 

+ 2\rj + _ Q \ exp [- (r s + r L ) t/2] cos [8m t- Arg(^) + Arg(»? + _)]} , 

where r(K s -> tz+tz-) is the decay rate of the short-lived state into tz+tz- and 
V+- = <ji + n-\T\L)l<ji + n-\T\S'}. The time dependence of the tz+tt decay rate 
in addition to the two terms due to Kg and K L decaying into tz + tz~ has an 
interference term showing the characteristic precession between the Kjr, and K s 



262 



C. Rubbia 



states due to the mass difference. This interference term can be observed 
over a relatively long time interval. In the case of a thick regenerator 
|g|> \v+-\ an d f° r ear ly times the term in h+_| 2 can be neglected and to 
an excellent approximation also r s -\- T L ~r s : 



dN 
dt 



(t)~r(K s ^+iz-y\ Q \* e xp[-r s t) 1 



2\rj._ 



V\ Q \*exp[-r s t] 



cos ($mt-\-oc) 



The statistical accuracy to which the decay rate can be determined at each t 
is also proportional to 1/V\q\ 2 exp[—F s t]. Therefore the error with which 
the interference term \r)+-\ cos(Sm + a) can be determined is the same at all 
values of t for which the approximation |?? + _| 2 ~0 is valid. In practice the 
time interval over which the interference can be observed is determined by 
the largest regeneration amplitude which can be realized, \Q/r) + _\~50 and by 
the contributions of systematic errors. 

The time-dependent interference pattern has been very extensively studied 
at CERN for carbon [7] and copper [8, 9] regenerators. A typical inter- 




/-1.0- 



10 11 12 13 TdO- ,0 sec) 



Fig. 1. - Time-dependent interference between the CP violating amplitude 
K x -> 71+tt;"- and the K s ->tc + tc-. The oscillation is showing the characteristic 
precession between the K L and K g phases due to the mass difference, o, ref. [14]; 

•, ref. [9]. 



The K^— Kg mass difference 263 

ference term is shown in Fig. 1. The precession between the K L and K s states 
due to the mass difference is beautifully demonstrated over a complete oscil- 
lation. From the frequency of the oscillation observed in the three investi- 
gations at CERN [7, 8, 9] the mass difference is measured to be : 

8m = (0.543 ± 0.016) x 10 10 s" 1 . 



4. - Experiments with two regenerators. The «gap» method and the « zero- 
cross » method. 

Another approach to the determination of the K^ — K s mass difference 
consists in using two sheets of matter. In the so-called « gap » method [10] 
the regenerated iz + -k~ intensity after the second one is determined as a function 
of their separation. In the « gap » method events are collected from the 
same decay volume for different regenerator geometries whereas the K^-Kg 
interference method is based on the observation of the decay rate from dif- 
ferent decay regions for a fixed geometry. This technique is less sensitive to 
variations of the detection efficiency over the decay volume but it requires 
an excellent monitor and substantial corrections due to interferences with 
the CP violating decay K^-^Tr+Ti - . 

The method can be easily understood expressing the tc+ti - decay amplitude 
at the proper time t from the exit face of the second regenerator as the sum 
of the contributions of the two slabs and of the component arising from 
K £ ->7r+7T~ decays. Let At be the time interval between the exit faces of 
the two slabs of regeneration amplitudes q x and q 2 , respectively. Then: 

<7r+7t-| T 1 1, A?> = [q x exp [— iM s {t + At)] + g 2 exp [— iM s t] • exp [— iM L At]] • 

• <ji+n-\T\Sy + exp [— iM L (t + At)](n+7i-\T\L) = 
= exp [iM L Af ]{[& exp [— i(M s — M L )kt] + q 2 ] exp [— iM s t] • 

• (7z+7i-\T\Sy + exp [— iM L t]<7i+7r\T\U)} . 

As At is varied for a constant t, the last two terms combine to give a fixed 
vector, whereas the first one decreases in magnitude and precesses around 
this fixed direction. The two pion decay rate can be easily calculated taking 
the square of the expression above: 

/(/, A0 = I x {t + A0 + I 2 {t) + iVljUt + A0/ 2 (0 cos (Sm A0- / , 
where : 

7 = r(K fl -^7c+7r-)-|i ?+ _|* f 



264 



C. Rubbia 



i- 
Z 
Z) 

O 
o 

O 



O 

2 



z 
o 

LU 



O 

DC 
LU 
Q. 

(/) 

z 

LU 

> 



11T- 



": 3 



DO 

Z 



t i r 



l i r 



MINIMUM X 1 SOLUTION 
USING THE EXPERIMENTAL 
VALUES OF F 21 . 



* 2 = 1.3 
(5 = ±.43 



ARG if - ARGti = ±40< 

21 '+- 



—S 



J I L 



J L 



4 6 8 

GAP (INCHES) 



10 



12 



Fig. 2. - Number of t+tt events as a function of the gap size and the best fit 
to these data (from ref. [11]). 



The Kr-K s mass difference 



265 



7 1 (r + A0 = /XK fi 



-^ + [- N 2 exp [-r s (t + Af)] + h + _| 2 + 

+ 2\Qin+-\ exp [— r s (/ + AO/2] cos (8m(' + AO + a)] , 



/ 2 (0 = r(K s -> TT+Tt-) [| & | 2 exp [- r s * ] + h + _| 2 + 

+ 2 1 £2*7+ -I ex P [— r s t/2 cos (Sm t + a)] . 

The terms / > 7^ and 7 2 have a very simple physical meaning. They are the 
decay rates one would observe for no regenerator, and only regenerators 
1 and 2, respectively. Therefore I x and I 2 have a form analogous to the one 
of the single regenerator experiments. 

Determinations of Sw with the « gap » method have been reported by the 
Princeton group [10, 11]. The results are shown in Fig. 2. The two pion 
decay rate has been observed as a function of the spacing between two regen- 
erators of the same material. The ratio q 2 jq x is then known. However the 
A/-dependence of the term proportional to cos(8raA0 and of J^f + Af) 
are not completely separated out in the fit and the result depends somewhat 
on the choice of the parameters like |?? + _|, r s , Arg(^ + _). The result of the 
most recent Princeton experiment [11] is $m/r s = (0.445 ± 0.038). 

In order to overcome these difficulties, a slightly different method, called 
the « zero-cross » method has been used by a CERN group [12]. These 
authors have determined experimentally both I(t, A?) and I , ^(f+Ar),/^), 



N,r N t- N z 



2VN, N 2 



DISTANCE BETWEEN BLOCKS l=529.9n 



Am=(0.538t00093)xl0*'" sec 



-v£ 



2VN, N 2 
\ 
\ 



\ 



GEOMETRY 2 
DISTANCE BETWEEN BLOCKS l=778.8n 



Am=(0.5<i6?0.0082)xl0 



5.0 xl0- ,u sec 



i\ 



PROPER TIME 



V 



\ T 
V 



Fig. 3. - « Zero-crossing » point of the interference between two slabs in the 
experiment of ref. [12]. The two curves correspond to different spacings be- 
tween the two blocks, a) Geometry 1 : distance between blocks is 529.9 mm. 
b) Geometry 2: distance between blocks is 777.8 mm. 



266 C. Rubbia 

owing to their very simple physical meanings. Then the precession due to 
the presence of A? can be evidenced directly from the experimentally observed 
rates, as follows: 

2V/ 1 (f + A0/ a (0 

In order to reduce the possibility of errors coming from incorrect monitoring, 
rather than varing the spacing A/ between the two slabs, different values 
of A? have been explored taking events of different kaon momenta p K , since 
At = A/-ra K //?K- The most sensitive region is around the value for which 
cos(SmA?)~0. For this reason the method is denominated « zero-cross ». 
Preliminary results from the CERN group [12] are shown in Fig. 3. The 
result is 

Sm = (0.542 ± 0.0060) x 10 10 s~\ 



5. - Concluding remarks. 

The recent experiments clearly demonstrate the effects of a very small 
but finite mass difference between the K L and K s states; when completed 
by the investigations of Piccioni and collaborators which have beautifully 
demonstrated that the the long-lived state is heavier [13] they give a precise 
determination of the mass difference. 

If the results can be combined with experiments which observe the inter- 
ference of K L and K s in the 7i + 7i~ decay channel from a state which is pre- 
ponderantly K (S= + 1) at the production one finds that [14, 15] 

Arg(7/ + _) = (41 ± 5)°. 

This result is in agreement with a class of theories which give Arg(?? + _) = 
= tan -1 (2 8m/r s ) c± 41°, and in particular with the « super- weak » for which 
such a prediction is exact. 



REFERENCES 



[1] M. Gell-Mann and A. Pais: Phys. Rev., 97, 1987 (1955). 

[2] J. S. Bell and J. Steinberger: Proc. Oxford Conf. on Elementary Particles (1965), 

p. 165. In the present paper we shall follow rather closely the formalism developed 

in this reference. 



The K L — Kg mass difference 267 

[3] L. B. Okun' and B. Pontecorvo: Z. Efejp. Teor. Fiz., 32, 1587 (1957). The argu- 
ment is valid only for P-conserving interactions; see S. L. Glashow: Phys. Rev. Lett., 
6, 196 (1961). 
[4] J. Christienson, J. Cronin, V. Fitch and R. Turlay: Phys. Rev. Lett., 13, 138 (1964). 
[5] A. Pais and O. Piccioni: Phys. Rev., 100, 1487 (1955). 
[6] M. L. Good: Phys. Rev., 106, 591 (1957); 110, 550 (1958). 
[7] M. Bott Bodenhausen, X. de Be Brouard, D. Cassel, D. Dekkers, R. Felst, 

M. Vivargent, T. Willits and K. Winter: Phys. Lett., 23, 277 (1966). 
[8] C. Alff-Steinberger, W. Heuer, K. Kleinknecht, C. Rubbia, A. Scribano, J. Stein- 

berger, M. Tannenbaum and K. Tittel: Phys. Lett., 21, 595 (1966). 

[9] P. Darriulat, H. Faissner, H. Foeth, C. Grosso, V. Kaftano, K. Kleinknecht, 

C. Rubbia, J. Sandweiss, A. Staude and K. Tittel: to be published in Phys. Lett. 

(1969). 

[10] J. Christienson, J. Cronin, V. Fitch and R. Turlay: Phys. Rev., 140, B 74 (1965). 

[11] R. K. Carnegie: Technical Report No. 44, Oct. 7, 1967, University of Princeton, 

Princeton, N. J. 
[12] H. Foeth, M. Holder, E. Radermacher and A. Staude; P. Darriulat, J. Deutsch, 
M. Cullen, K. Kleinknecht, C. Rubbia and K. Tittel; C. Grosso and M. Scire: 
to be published in Phys. Lett. (1969). 
[13] R. Good, W. Melhop, O. Piccioni, R. Swanson, S. Murty, T. Burnett and C. Hol- 
land: Phys. Rev. Lett. 
[14] A. Bohem, P. Darriulat, C. Grosso, V. Kaftanov, K. Kleinknecht, H. Lynch, 

C. Rubbia, H. Ticho and K. Tittel: Nucl. Phys., B 9, 606 (1969). 
[15] D. A. Jensen, S. H. Aronson, R. D. Ehrlich, D. Fridberger, C. Nissim-Sabbat, 
V. Telegdi, H. Goldberg and J. Salomon: Phys. Rev. Lett., 23, 615 (1969). 



Suggestion for a More Precise Measurement 
of the t)+- Phase. 



J. Steinberger 

CERN - Geneva 
Columbia University - New York 



This volume represent an effort, on the part of Professor Amaldi's friends 
and colleagues, to show some token of their affection and respect. In the 
case of this particular contribution, it is unfortunately a very small token. 
Quite apart from personal limitations, experimental particle physics moves 
slowly and requires the collaboration of many, so that it is rarely possible 
to produce a result for an occasion. I am in this way constrained to offer 
a mere suggestion for an experiment. It is done with mixed feelings : normally, 
I would prefer to wait for the result, but for the occasion I would like to make 
some contribution, even if it must be incomplete. I hope that it will be judged 
in this way. 

It is some years since the discovery of CP violation, but despite substantial 
effort, it has been observed only in the K° system. It is possible that we will 
continue to be restricted to the K° in the future as well, in our efforts to learn 
about CP violation. It is then fortunate that several CP violating para- 
meters in K° decay, and in particular 

n _ <+-\r\K£> 
V+ -~<+-\*\Ks>' 

the relative amplitudes of long- and short-lived transitions to the charged 
pion state, can be measured with precision. In this note I point out a variation 
in the present line of these experiments which should permit an improvement 
in the measurement of the phase of t] + _ . The same technique is in principle 
also applicable to the phase of the corresponding neutral decay amplitude 
ratio r] 00 , but the experimental difficulties in7r°7T: decay are somewhat greater, 
so that it will be some time before the method can be expected to be useful 
also here. 



Suggestion for a more precise measurement of the rj + _ phase 269 

The ?? + _ phase can be measured by observing the time dependence of the 
tc + 7t:- decay of a kaon state which is given at time x = 0: *F(0) = q\S} + |L>. 
This time distribution has the form: 

(1) U) = H 2 exp [-/>] + |*? + _| 2 exp [-T L x] + 

+ 2|^ + _| exp [— Fx ] cos (Am r — q^J . 

Here F s and 7^, are the short- and long-lived widths, respectively; F is 
(F s + F L )/2 ; x is the time in the K° rest frame ; and Am = m h — m s . 

I + Jx) is plotted in Fig. 1 for a state with q = 1 (V(0) = |^°>). It is 
experimentally possible to measure this time distribution accurately, except 
in the short time region (shaded in Fig. 1), which is inaccessible for shielding 
reasons. Unfortunately, this does not yield a corresponding precision in y n : 
Since the <p n dependent interference term is strongest at x = 10 fr s , what 
is measured best is the quantity 10 Am/-T 5 — 9V-* A precise determination 
of 9 9 » ?+ _ is only possible if a correspondingly precise value of Am is available. 
To be completely explicit, if the error in cp n+ _ is to be less than some number, 
say A99, the uncertainty in the mass difference must be less than A(Am/.r s ) ^ 
A99/IO. The measurement of the mass difference with this precision turns 
out to be the bigger experimental problem. 

It is the main contribution of this note to point out that the mass dif- 
ference can be measured with the same apparatus as the 7r+7r~ interference 
term, and simultaneously. 

The suggestion is to measure, simultaneously with the -k+tt decay rate, 
also the charge asymmetry in the lepton decay as a function of x. This charge 
asymmetry is also governed by an interference between K s and K L am- 
plitudes; the dominant term has in fact the same form as the interface term 
in 271 decay. 

If 

_ (7c~e + v\r\K} _ AS= A.Q violating amplitude 



<7i e + v\x\K} AS— AQ conserving amplitude 
and if CPT is assumed, then 

(2) (3(7) ^+""^- ^ 

^ 2(1 — |x| 2 ) [(exp [— r s r] + exp [— T z t]) Ree + exp [— Fx] cos Amr] 
= |1 + x| 2 exp[— r s x] + |1— x| 2 exp [— T^t] — 4 Im x exp [— Fx] sin Am ' 

where 7V + and N_ are the decay rates to positive and negative leptons, respec- 



270 




x r s 

Fig. 1. - Two pion intensity and leptonic decay asymmetry as a function of 
the time in the K° rest frame. 



tively, and e is the parameter in the expansion of K L and K s in terms of 
K and K: 



\S> = 



1 



[(l+ fi )|tf> + (l-e)|*>]. 



V2(l + |e|) 2 
CPT is assumed, and higher order terms in e have been omitted. 



Suggestion for a more precise measurement of the t] + _ phase 271 

Expression (2) is also shown in Fig. 1. At long times expression (2) 
reduces to the already observed small (~2.7xl0~ 3 ) asymmetry of K. z : 

1 — Ixl 2 
^T-*oo)s S d i = 2Rec-y r -£L 

At short times the term exp [ — rr] co s Amr dominates and at time r = 0, 

It may be useful here to point out an important feature of this approach. 
Both the 2jc distribution (1) and the asymmetry distribution (2) are slightly 
modified in an experimental situation, due to several small effects, such as 
the propagation of the kaon in the target in which it is produced, and the 
scattering of the kaon on the collimators and y-ray filters which are com- 
monly introduced. The main effect of this on the measurement of (p n and 
Am is to introduce a small phase change <p into both expressions (1) and (2), 
so that the arguments of the cosine function in the interference terms are 
changed to Amr — (pt, + _-\-<p and Amr + 99 , respectively. <p , in an experi- 
ment currently planned, is of the order of (1 -=-2)°. This is somewhat greater 
than the error which is anticipated. However, cp is common to expressions 
(1) and (2) and if we think of the charge asymmetry measurement as a deter- 
mination of the quantity Amr -\-<p , we can see that in the comparison of 
the two distributions cp n can be extracted without separate knowledge of 
Am or 9? : This, however, is only a heuristic way of understanding the manner 
in which the errors enter. In practice it will be necessary to analyse the two 
experiments to find Am and cp . However, the error in AmT+9? for r~lQ/r s 
will be smaller than the error in q> Q or in Amr. 

It is necessary to discuss here a problem in connection with the measure- 
ment of the rest-frame time r in the leptonic decay. Let t = dmjcp, where 
d is the (measured) distance before decay, and m and p are kaon mass and 
momentum, respectively. The difficulty is that p is not directly measurable 
since the neutrino is not observed. It is however possible to proceed as follows. 
For each event the directions and momenta of the two charged particles 
are measured. We can then define p' = \p + + p~-\ (*) and t' = dm/cp', and 
tabulate the experimental asymmetry as a function of t'. It is then necessary 
to fold the transformation t<->V into the expression (2), a process which 



(*) Other definitions of p' , which serve equally well or perhaps even slightly better, 
are possible. This definition serves, however, to illustrate the method. 



272 J. Steinberger 

requires a knowledge of the geometry of the apparatus and the beam mo- 
mentum distribution. Both of these can be known with sufficient accuracy 
so that this step need not necessarily increase the error appreciably. 

In the remainder of this paper we will discuss the precision which may 
be obtained in y n in a particular experiment. In a proposed experiment, the 
kaons are produced at 75 mrad to a 24 GeV/c proton beam. The 'detector 
is assumed to have a sensitive decay region between 2.2 and 11.5 m from the 
point of kaon production. The main contribution to the result will be from 
kaons between 6 and 12 GeV/c, exploring the r interval 4 < r s r < 30. It is 
expected that 1.5 xlO 7 leptonic decays and 5xl0 4 K L ^Tz+it- decays per 
short-lived lifetime can be accumulated in an extended experiment. The 
theoretical expressions (1) and (2) are modified to account for the fact that 
both K and K are produced at the target. This has the effect of diminishing 
the magnitude of the interference term by the factor 



/k+V 



Experimentally it is known that % varies from 0.5 at 6 GeV/c to 0.85 at 12 GeV/c 
for the postulated conditions. 

Calculations have been performed which consist of « generating » experi- 
mental data according to expressions (1) and (2), using the present experi- 
mental values for Am, r s , rj, cp n , %, etc., and then in turn taking these « data » 
and inverting the analysis to find the same parameters. In this latter part 
X and <p Q are left as functions of the momentum. The result is that the stati- 
stical error for cp n is expected to be ~f°. 

Of course there may be unanticipated problems in the successful exploitation 
of this suggestion, but the experiment is in process of preparation at CERN, 
and in a year or two should be completed. 



I wish to thank Drs. K. Kleinknecht and P. Steffen for discussions and 
the calculations referred to in the last paragraphs of this note. 



An Amateur's View (*) of Particle Physics (**). 



V. F. Weisskopf 

M.I.T. - Cambridge, Mass. 



Here are some impressions of a non-expert on the present state of par- 
ticle physics. 



1. - The third spectroscopy. 

One of the most striking aspects in the recent development is the ever- 
growing list of excited states of the bar yon and meson. Remember the fact 
that nine years ago the only states known were p, n, J\P*(1236), 2, A, S, 
and 7T, K. Compare this with today's Rosenfeld table. The list is increasing, 
more levels are found every year and their quantum numbers become better 
known. It is a slow and painful task indeed; each simple level requiring 
many man-years of work. It is not made easier by the fact that, often, the 
widths of the levels are comparable to the level distances. This brings in a 
slightly disturbing feature: Some baryon levels are established only by 
phase analysis of meson scattering; they do not appear as a bump over a 
background in a scattering experiment. Such things happen very rarely in 
atomic or nuclear spectroscopies. 

A relatively recent innovation is the production of certain mesons by 
electron-positron collisions. This is an unusually clean way of producing a 
single vector meson — only that type can be produced singly — free of nearby 
sources of strong interaction ; it allows a better determination of the relevant 
properties. 



(*) Amateur: 1) A person who does something for the pleasure of it rather than 
for money. 2) A person who does something more or less unskilfully. (Webster's New 
World Dictionary.) 

(**) This work has been supported in part by the Atomic Energy Commission under 
Contract Number AT (30-1) 2098. This article is an enlarged version of a similar paper 
which appeared in Comments on Nucl. and Particle Physics, III, 1 (1969). 

18 



274 V. F. Weisskopf 

There are a few immediately obvious features in the level structure of 
hadrons. In contrast to atomic and nuclear spectra the level distances are 
comparable to the mass of the object. A striking feature is the existence of 
practically degenerate isotopic multiplets, another is the restriction of strange- 
ness to negative values S> — 3 for baryons and to |S|< 1 for mesons; finally 
there is a connection between the isotopic spin and the strangeness quantum 
numbers. This has led to the classification of levels by SU 3 and the discovery 
of SU 3 supermultiplets. The newly discovered levels seem to fit resonably 
well into this scheme. 

For me, the SU 3 classification is based on the quark model, with three 
types of quarks, which hadrons are composed of baryons being three quarks 
and mesons being quark-antiquark pairs. Thus, mesons, but not baryons, 
can be singly created and destroyed. The trichotomy of quark types is repre- 
sented by a formalism with three « unitary spins » : isotopic, w-spin, and u-spin, 
each representing the alternatives given by one pair of the three types, just 
as the ordinary spin represents the alternative of spin-up and spin-down. 
Isospin and strangeness of hadrons are directly obtained by summing the 
corresponding quantities of the quarks; this simple rule explains the actual 
relations and restrictions among these quantum numbers. Indeed, one finds 
only the angular momentum quantum number to be unrestricted ; a reflection 
of internal orbital excitations of the quarks which gives rise to families of 
hadrons with equal intrinsic quantum numbers but different angular mo- 
menta. 

If one assumes that the binding force which keeps the quarks together 
is independent of the isotopic spin, and weakly dependent on the other spins, 
the main features of the baryon and meson spectrum can be reproduced. 
The systematics of quantum numbers, the multiplet structure, some features 
of transition probabilities between levels, and ratios between decay rates of 
mesons fall into place. 

Of course, quarks have never been observed. Grave problems arise if 
that model is taken too seriously. However, it serves as a simply describable 
realization of SU 3 symmetry. The latter is what remains of the quark model, 
if one removes the quarks — the grin of the Cheshire cat. Why it works, is 
still one of the great miracles. 

Whether the quark-idea is correct or not — it is improbable that it will 
turn out to be correct in its present simple form — the experimental evidence 
of hadron spectra points to an internal structure of the nucleons and mesons. 
In some ways the situation is reminiscent of atomic and molecular physics 
before Rutherford and Bohr. We know much of the spectrum of the nucleon, 
we know something about the force between nucleons; it is a relatively com- 
plicated force, attractive at larger distances, repulsive at smaller distances, 



An amateur's view of particle physics 275 

spin and symmetry dependent. In this respect it is similar to the chemical 
force between atoms. Only after the atomic structure was elucidated, one 
found out that the chemical force is based upon the more fundamental electric 
force between atomic constituents. It may well turn out that the nuclear 
force also will be understood as a consequence of more fundamental inter- 
actions and processes within the nucleon. 

It would be misleading, however, to overstress the analogy between atomic 
structure and nucleon structure. As it was remarked before, the excitation 
energies of the nucleon are of the order of its rest mass energy, a circumstance 
which introduces new and badly understood features. With the present 
mathematical techniques, we are not able to deal with composite systems 
where the interaction is so strong that binding energies become comparable 
to the rest mass. One of the consequences of this situation is the important 
role of virtual particle pairs ; such a system becomes an agglomeration of pairs 
of particles and antiparticles ; the number of constituents, as it were, is always 
large and indefinite. It is to be hoped that methods can be found to deal 
with such conditions. 



2. - Electron scattering. 

The growing availability of high-energy electron beams is noticeable in 
the increasing number of interesting experiments with electrons. The resulting 
elastic form factors (electric and magnetic) of the nucleon are not yet under- 
stood. Their dependence on the momentum transfer q goes as g~ 4 at high 
^-values. The fact that the form factor decreases smoothly to zero at high 
#'s indicates that the nucleon is an extended system and that there is no 
hard and small core noticeable at the center of the nucleon. It does not 
exclude the possibility that the nucleon is made of hard and small constituents. 
It only shows that the charge and magnetic distribution has no accumulation 
at the center as it has in the hydrogen atom. 

The inelastic electron scattering has turned out to be most interesting. 
The excitation of higher baryon states by this process is a repetition of the 
Franck-Hertz experiment, more than 50 years later at a billion times larger 
energy scale. The form factor of these excitations seems to have a similar 
^-dependence as the elastic one — no wonder, since we expect the excited 
states to have similar charge and magnetic distributions. An interesting feature 
appears when one looks at the very large energy transfers of the scattered 
electrons, way above the known resonances. Then the strong ^-dependence 
disappears and the scattering seems to be independent of the momentum 
transfer, apart from the trivial electric charge effects (Mott scattering). A 



276 V. F. Weisskopf 

lack of ^-dependence indicates a scattering object smaller than the length 
associated with the momentum transfer. Did we hit here some very small 
entity or entities within the nucleon, some constituent or perhaps the quark? 
Whatever the detailed interpretation of these results may be, the absence of 
a ^-dependence at high-energy losses certainly indicates the existence of a 
length, small compared to 10~ 14 cm, which should play an important role 
within the nucleon. 



3. - Current algebra. 

How can we look into the dynamic situation inside the hadrons ? Strong 
interaction processes seriously distort the hadron under observation. Weak 
and electromagnetic processes, however, leave it intact and can be directly 
interpreted as the effect of currents inside the hadron. The most obvious 
example is the electromagnetic current density j% M (four vector) whose matrix 
elements determine, and can be determined by, electromagnetic phenomena. 
Similarly there exist four more current densities which, in the same way, 
determine the weak interaction phenomena. Why four? Firstly, the weak 
interaction as exemplified by lepton-pair emission, has two realizations, 
non-relativistically speaking, the spins of the leptons may be parallel or anti- 
parallel. This is connected with the fact that any weak process is determined 
by two currents, /£ and j^ , the vector and the axial current. In addition, 
there are two types of weak processes: strangeness conserving and strange- 
ness changing (JA^J = 0, 1), which leads to two more currents: j%, u j^. 
Ideally one could measure all matrix elements of the five current densities, 
if every conceivable electromagnetic or weak process were entirely known. 
So far, we known very little about them. One most important known fact is 
the discovery (Gershtein-Zeldowitch, Feynman-Gell-Mann) that y'^ apart 
from a constant, is an isotopic brother of j^ M , that is, it differs only by the 
fact that the charge changes in the former. (This difference is a rotation in 
the isotopic spin space.) Current algebra is a bold generalization of this 
idea, which assumes that all five of them are brothers, they are supposed 
to belong to a family of sixteen current densities which I will now describe. 
Again we make use of the quark model. Whatever the dynamics are, there 
are currents running in this model; for example, we can define three cur- 
rents which describe the flow of each one of the three components of the 
isotopic spin. Since there are two more unitary spins, the £/-spin and the 
F-spin, one would think that there will be nine currents. However, the three 
types of spin are not independent — an /-flip followed by a U- and a F-flip 
brings us back to the original state; therefore, there are, in fact eight 



An amateur's view of particle physics 277 

independent current densities: j%, X = 1, ..., 8. All of them are vector fields. 
The electromagnetic current is included and they are all conserved currents. 

The second half of the family is obtained by considering something 
which — non-relativistically— corresponds to the spin density (ordinary angu- 
lar momentum spin) of the quarks. It represents an axial vector field within 
the hadron. If we associate the angular momentum spin density of the 
quark (in this non-relativistic approximation) with its unitary spin com- 
ponents (/-, U-, F-spin), we obtain eight axial current densities jf, X = 1, ..., 8. 
They are not conserved currents since a « spin density » is not a conserved 
vector field. 

The weak currents are supposed to be some linear combinations of the 
sixteen currents, namely the ones that give rise to the relevant charge and 
strangeness change. 

What follows from this assumption? Evidently the Feynman-Gell-Mann 
relations between the electric current and the vector part of the weak current 
is part of this assumption. But there are more; there exist simple relations 
between those sixteen currents since, in terms of the quark model, they can 
be represented by simple operators. For example, equal-time commutation 
relations exist, such as 

where f XvK are constants and the zero index refers to the time components of 
the current densities. This is equivalent to a sum rule 

2(a\M\b)(b\N\a)-2(a\N\b)(b\M\a) = const (a|P|a) , 

b b 

where M, N, P, are operators connected with the currents and a and b are 
quantum states. One example [1] refers to neutrino cross-sections: 



lim 



da(v P ) ^v)l = 5! (c08 . (?c+28in . (?Q)f 



dk 2 dk* 



where a{vp) is the total neutrino cross-section of a proton, k is the momentum 
transfer, G the weak coupling constant, and 6c, the Cabibbo angle. We are 
far from able to test such relations. 

Current algebra proper establishes connections between currents, but says 
nothing about the currents themselves. There are approximate ways to get 
some limited information about the current distribution within the hadrons. 
A baryon, for example, can be considered as surrounded by virtual meson 
fields. The mesons with lower mass contribute more strongly and at larger 



278 V. F. Weisskopf 

distances compared to those with higher mass. Thus particularly the pions, 
but also the kaons and vector mesons such as the p or w will play a major 
role in the « meson cloud ». Therefore, it is plausible that the vector and 
axial vector current densities have some relations to those meson fields 
which have similar geometrical properties. For example, the vector current 
densities, such as the electromagnetic one, should be related to the p, co, 
cp fields ; a suitable axial vector current density (the one that carries isotopic 
spin) should be related to the pion field, since the divergence of the axial 
current density is a pseudo-scalar field, like the pion field. Such assumptions 
are known under the name of « vector meson dominance » and « pion domi- 
nance ». The former is equivalent to an assumption that for small momentum 
transfers— only for those will this restriction to the lowest mesons hold — a 
light quantum interacts with a baryon in a similar way as a vector meson 
would, apart from a proportionality constant. That constant is determined 
by the electromagnetic properties of the vector mesons, and can be deduced 
from the decay probabilities of these mesons into electron pairs. This assump- 
tion has proved to be quite useful for the prediction of photoprocesses 
with baryons. 

The relation between the axial current density and the pion field has 
many interesting consequences. It is assumed that the divergence of the axial 
current density is proportional to the pion field. The proportionality constant 
is given directly by the pion decay into a lepton pair and it represents in 
some way, the « axial charge » of the pion. This assumption (together with 
some assumptions about a reasonable behavior of the matrix elements of 
the pion field) gives rise to a connection between the « axial charge » (the 
pion-nucleon coupling constant), and the axial weak interaction coupling 
constant of the nucleon (Goldberger-Treiman relation). The connection 
comes about in the following way: The weak interaction of the nucleon is 
caused by the axial current, which is proportional to the pion field which, 
in turn, is coupled to the nucleon and therefore dependent on the pion- 
nucleon coupling constant. 

The Adler-Weisberger relation is another example which can be derived 
from this connection between axial weak interaction and pion-nucleon coupling. 
Here one also uses current algebra which establishes a connection between 
axial currents and vector currents. One then gets an expression for the 
ratio of the axial — to the vector coupling of weak interactions, in terms of 
meson-nucleon cross-sections. 

The relation between the axial current and the pion density has an inter- 
esting bearing on the question of the conservation of the axial current. The 
divergence of the axial current density — which should vanish if the current 
were conserved — was assumed to be proportional to the pion field. The 



An amateur 's view of particle physics 279 

pion field in the vicinity of a hadron is spread out over relatively large dis- 
tances of the order of m" 1 . If this spread is reasonably smooth, one would 
conclude that the Fourier components for wave numbers much higher than 
m n should be small. Hence, matrix elements of the divergence of the axial 
current density will be very small for momentum transfers larger than m n . 
This means that the axial current is conserved in this limit. The Partial Con- 
servation of Axial Currents (PCAC) gives rise to a number of simple relations 
for the interaction of soft mesons with hadrons. In electrodynamics where 
the coupling is also mediated by a conserved current, absorption, emission 
and scattering of long wave length light quanta is given by simple expressions 
proportional to a power of the electric charge. We find similar approximate 
expressions involving the axial charge for the corresponding pion processes. 



4. - Strong interaction processes. 

The theoretical description of the interaction between hadrons is a more 
complex problem than that of the interaction of leptons or the electromagnetic 
field with a hadron. The concept of current densities is adapted to the case 
where the interaction is weak and the interacting field is not much distorted 
near the hadron. This is the case when one can use perturbation theory 
and the first approximation of an interaction is the dominant one. Then the 
interacting field is coupled to those features of the hadron which exhibit 
the symmetries of the unperturbed field. When hadrons interact among 
each other, this approximation method is no longer applicable since the inter- 
actions are strong. Yet it remains to be seen whether the current concept 
can be sufficiently generalized and adapted to problems of strong inter- 
actions. Perhaps attempts at using a so-called phenomenological Hamilto- 
nian are efforts in that direction. 

Most of the work in hadron interactions is based on a different approach. 
The scattering amplitude A of one hadron scattered by another is a function 
of the relative energy and the momentum transfer. Since there is no theory 
for the calculation of this magnitude, one establishes some general rules to 
which the amplitude A is subjected, and then one tries to extract some 
theoretical predictions which may follow. The general rules are based on the 
following four items: 

a) relativistic invariance; 

b) causality; 

c) unitarity; 

d) analyticity. 



280 V. F. Weisskopf 

Relativistic quantum mechanics tells us how to describe unambiguously 
a group of non-interacting particles with definite masses and spins. Every 
scattering process begins and ends with such a group. It, therefore, defines 
the variables on which the scattering amplitude depends. Items b) and c) 
establish relations which the scattering amplitudes must fulfill, such as dis- 
persion relations and the optical theorem. Item d) contains the assumption 
that the scattering amplitude is an analytic function of the relevant variables 
apart from certain singularities which have well-defined physical significance. 
I suppose that any imaginable and resonable theory will always give rise to 
such functions with certain poles, cuts, and definite asymptotic properties. 

The importance of item d) comes from two circumstances : First, a pole 
of the scattering amplitude represents a stationary or metastable state of the 
composite system of the two scatterers ; the relative energy at which the pole 
occurs is the mass of the stationary state and the residue is connected with 
the coupling constant of the binding force. Some of these poles may have 
an overriding influence on the energy dependence of A, which sometimes 
may lead to a simple expression for A involving only a few dominant poles. 
The second point is the crossing relations: There is a relation between the 
amplitudes of two reactions which differ by replacing a particle coming in 
by its antiparticle going out, and also replacing an outgoing particle by its 
antiparticle coming in. The amplitude of one reaction is an analytical con- 
tinuation of the other in certain variables. If we know something about 
one reaction, we can arrive at some conclusions about the other; for example, 
a strong pole in one channel may have a noticeable effects in the other. Here 
again the fact that the relevant energies are of the order of the rest masses 
of the particles involved plays an important role. Therefore a pole in one 
reaction channel may not be so much further away from an interesting energy 
region in the crossed reaction, than it is from a relevant energy region in its 
own channel. This consideration also shows why crossing relations are rela- 
tively unimportant in nuclear or atomic physics, where the relevant energies 
are small compared to the rest masses. 

The most important experimental fact of hadron collisions is the following 
one : In the case of elastic scattering and of those inelastic scatterings where 
the quantum numbers of the incident and outgoing particles are the same, the 
cross-sections seem to reach an energy independent asymptotic values at 
high energies. For inelastic processes, where quantum numbers are exchanged, 
the cross-sections vanish as a negative power of the energy. This experimental 
result is not yet explained by any theory, but it is often used as a basis of 
deriving other results. 

A number of conclusions can be derived from this observed asymptotic 
behavior, with the help of the items a)-d). One is the Pomeranchuk theorem, 



An amateur's view of particle physics 281 

concerning the quality of particle and antiparticle elastic scattering {e.g., 
7T++P and 7T-+P at high energies). Other recent conclusions are the « finite 
energy sum rules ». If one assumes — in some cases there are good reasons — 
that the difference between the actual and the asymptotic amplitude of a 
reaction vanishes stronger than a certain power of the energy (stronger than s _1 ), 
the integral of the actual amplitude over the energy from zero up to a finite 
energy E is essentially the same as the integral of the asymptotic amplitude 
over the same energy range. The upper limit E must be an energy at which 
the asymptotic form is already applicable. This relation establishes an inter- 
esting connection between the low energy behavior governed by resonances, 
and the high-energy behavior which sometimes can be predicted by the 
Regge-pole theory discussed later. 

One of the important concepts in describing interactions between 
hadrons is the concept of exchange. In every two-body reaction something 
is transferred frome one partner to the other: It is momentum only in an 
elastic reaction, it can be any intrinsic quantum number in others. (We call 
« intrinsic » any quantum number except angular momentum.) It is some- 
times useful to describe the reaction by asserting that a suitable particle is 
exchanged, which is coupled by an interaction with both partners and carries 
over the quantities. The situation could be described in the following way: 
A hadron is considered as a conglomerate of hadron pairs whose quantum 
numbers add up to the intrinsic quantum numbers of the hadron under con- 
sideration — a proton is a conglomerate of n7i + , K + A, etc. When hadrons 
meet they may exchange a member of those pairs. The rules of these exchanges 
are hard to get at, since particles in the conglomerate do not behave like free 
particles. One must find means how to deal with this situation. The Regge- 
pole method is such an attempt: One makes use of the crossing relations 
and starts from another reaction channel (^-channel) in which the exchanged 
particle appears under more « natural » conditions, that is as a free particle. 
It is the channel where it is created by the fusion of two hadrons (more exactly, 
by the fusion of one of the incoming hadrons and the antiparticle of an 
outgoing one). In this channel the exchanged particle appears as a composite 
system made up by the two hadrons. Such a system may assume a number 
of states of different angular momentum each of which may be identified 
with some observed particle (Regge families). Each member of this family 
has the same intrinsic quantum numbers and could serve as an exchange 
particle in the reaction. In terms of a continuous /-variable, a Regge family 
is represented as a function m(J), where the actual particle masses are the 
values of m at integer (or half integer) /'s (Regge trajectory). 

The existence of these families or trajectories is exploited to express the 
scattering amplitude in terms of a sum of contributions from special poles 



282 V. F. Weisskopf 

(Regge poles) which appear if the angular momentum is used as a continuous 
variable /. Each Regge pole represents not only a single state of the com- 
posite system, but it encompasses the effects of a whole Regge family upon 
the scattering amplitude. It is assumed that these Regge-pole contributions 
have an overriding influence so that they dominate the scattering amplitude 
not only in the ^-channel, but also in the actual scattering channel. This may 
be an appropriate way of dealing with the concept of exchange in strong inter- 
actions, which takes into account not only the effect of exchange of one 
particle, but of each member of a Regge family. Its main results are state- 
ments about the scattering amplitude in the limit of small momentum transfer 
and high-energy. Hence, it is relevant for the asymptotic energy behaviour 
discussed before. 

It must be said that these Regge extrapolations from one channel to 
another introduce a number of arbitrary functions into the picture such as 
the « coupling strength » of the Regge trajectories (residues of the Regge 
poles). In order to satisfy the analyticity of A, these arbitrary functions 
must fulfill a lot of complicated conditions (evasions and conspiracies), and 
new « daughter » trajectories must be introduced which may not necessarily 
give rise to observed particles. In view of all these complications, together 
with the introduction of Regge-cuts — the same kind of treatment for the 
exchange of pairs of particles — one may ask whether the Regge-pole method 
provides deeper insight into what is really going on. I quote Van Hove [2] 
« The Regge-pole model is not a theory with a high predictive power, but a 
refined framework to correlate collisions — especially of inelastic type — to 
exchange processes ». 

Recently Veneziano has proposed an expression for the scattering am- 
plitude for the case of meson-meson scattering which, perhaps, goes a little 
further than the Regge-pole model. It is a simple expression representing a 
specific case of a Regge-pole model, in which all Regge trajectories and the 
functions describing their coupling strength are well defined. The trajectories 
are straight lines (m 2 is a linear function of /) and the fact that only mesons 
are involved, introduces symmetries which define all the coupling strength 
functions in an unambiguous way. The simplicity and the internal consist- 
ency of this expression are impressive. It seems that some of the conclu- 
sions one can draw from this expression are in approximate agreement with 
the present experimental knowledge on meson-meson scattering. Obviously, 
the Veneziano expression can only be a first approximation to reality; there 
are theoretical and experimental reasons for this. The lack of unitarity is 
of the former kind, the approximate nature of the results is of the latter. 
It is important to keep in mind that the Veneziano expression is not a theory 
of meson interaction. It is a guess at a result, in the form of a simple math- 



An amateur's view of particle physics 283 

ematical example for a scattering amplitude which fulfils most of the con- 
ditions that are imposed. At worst, it is a simple mathematical example of 
a scattering amplitude fulfilling most fundamental requirements; at best, 
it may be a first approximation to reality and may have the value which the 
Balmer formula had in the development of our knowledge of the hydrogen 
atom. 



5. - CP-vioIation. 

After the discovery of parity violation in 1957, Pauli exclaimed in a famous 
letter: « God is left-handed ». Actually, it turned out soon, that this is 
not so. God is right-handed in the antiworld where parity is violated in 
the opposite sense. However, seven years later we had reasons to feel uneasy ; 
Christenson and Cronin, Fitch, and Turlay found out that world and anti- 
world are not equivalent even if left is replaced by right. In other words, 
nature is not CP invariant. (C transforms world in antiworld, P left into 
right.) 

In order to find extremely small differences between a particle and its anti- 
particle, one must set up beat frequencies. That means, one must be able 
to create a linear combination of the two with well-defined phases. This 
can only, be done if there is an interaction which can transform one into 
the other. No baryon-antibaryon pair would do, since all interactions con- 
serve baryon numbers. No charged meson pair would do because all inter- 
actions conserve charge. An uncharged meson with strangeness zero is its 
own antiparticle so that it cannot be a candidate for particle-antiparticle 
mixing. All that is left are uncharged mesons with a strangeness different 
from zero. Weak interactions can transform one into the other since they 
do not conserve strangeness. Hence, the (K , K ) pair rose to such great 
fame. 

Thus K and K are related by the transformation CP: CP K = ~ K . As 
long as the weak interaction which transforms K into K is CP-invariant, 
the two states are equivalent and two linear combinations with slightly dif- 
ferent energy are formed, a symmetric and an antisymmetric one: 

-^i — ^o — ^o » K 2 = K -\- K . 

Since the mixing of K and K is done by a second-order weak process the 
splitting is extremely small; it is of the order of 10~ 5 eV. It is instructive 
to think of the analogous situation with two identical coupled pendulae. 



284 V. F. Weisskopf 

(This analogy was used by F. Crawford in a colloquium talk.) Such coupling 
also produces a symmetric and an antisymmetric proper vibration with 
slightly different frequencies. Usually the antisymmetric one has more friction 
than the symmetric one so that, after some time, the latter one only survives. 
The same is true in the (K , K ) system. Friction corresponds to weak decays. 
We direct our attention mainly to decay into pions, either into two, or into 
three pions, of the total charge zero. Two pions are even under CP-trans- 
formation, three pions are predominately odd if they emerge from a small 
source. Hence, the symmetric combination K 2 (which is odd under CP) 
can only decay into three pions, the antisymmetric K ± decays mostly into 
two. Since two pions have more phase space, the K x mode has more « friction » 
and the K 2 mode survives longer. 

If the weak interactions were not CP-invariant, K and K would no 
longer be equivalent in this interaction. This would have two consequences : 

a) the eigenstates, which we call K s and K L (short-lived and long-lived), 
would no longer be either symmetric or antisymmetric. They would be some 
other linear combination: K s =£ K x , K L ^K % . The same would happen in 
the case of two coupled pendulae, if the two pendulae were not exactly equal. 

b) The number of decay pions would no longer be an indication of the sym- 
metry of the decaying state; K 2 would also be allowed to emit two pions. 
As the result of both consequences, one would observe that the long-lived 
eigenstate sometimes emits two pions and not always three. This is what 
Christenson et ah, have found to our surprise and bewilderment, and they 
have shown that K and K are not equivalent ; world and antiworld are dis- 
tinguishable. This simplest situation is realized if only consequence a) of 
CP-violation occurs. This would be the case with a so-called « super-weak » 
interaction, which violates CP but is very much weaker than the ordinary 
weak interaction. Hence, it would practically not influence the ordinary 
weak decay into two or three pions, so that this alternative can still be used 
as a distinction between symmetry and antisymmetry. But, because of the 
extremely small split between the two modes — second-order weak inter- 
action — a very small CP-violating term (of the size of the second-order of 
weak interaction), would destroy the pure symmetry or antisymmetry of the 
modes. The long-lived mode would not be purely symmetric but would have 
a small antisymmetric admixture, which shows up in the form of a two pion 
decay. There is an indication that consequence b) may in fact not be present : 
The (7c+TT"/7c 7t )-ratio seems to be the same in the short-lived and in the long- 
lived, two-pion decay. This fact does not exclude consequence b) but if it 
were not so, one would be sure that b) is present. 

There is new independent evidence regarding the two modes of the 
(K , K ) system, which came from the comparison of the following two 



An amateur's view of particle physics 285 

decays of the long-lived mode: 

K i -^7T++e-+v, K i ->7i-+e++v. 

If K L were exactly symmetric in K and K , the two decays would be exactly 
equally strong. Actually, a group at Stanford and at Columbia [3] have 
found a difference of the order of one percent, a direct manifestation of an 
inequality of world and antiworld in the K^-mode. 

6. - General situation. 

It is questionable, whether our present understanding of high-energy 
phenomena is commensurate to the intellectual effort directed at their inter- 
pretation. We are able to describe the phenomena in terms of particle fields, 
scattering amplitudes, current densities, etc., by using a language which im- 
plicitly assume that there exists a valid quantum theory of particles with inter- 
actions — strong, electromagnetic, and weak — which explains everything in a 
consistent way along the accepted principles of quantum field theory. The 
experiments do not reveal any inconsistency with these principles. On the 
other hand, no such theory has yet been formulated. It is impossible to 
decide at this stage, whether this lack of success is caused by the mathematical 
difficulties of a field theory with strong interaction, or by the fact that we 
have not yet found the conceptual framework necessary to understand the 
situation. 

The present theoretical activities are attempts to get something from 
almost nothing; attempts to conclude as much as possible from a few general 
restrictions of the formalism, such as relativity, causality, unitarity, and 
analyticity. It is astonishing and exhilarating how much, in fact, one can 
conclude from so little input. It is a veritable « bootstrap » operation. But 
does it lead to a deeper understanding of what is going on within the hadrons ? 
Does it give us any insight into what lies behind that wealth of experimental 
material on resonances and reaction rates? It is true that the present for- 
mulation of weak interaction phenomena is impressive in its eloquent sim- 
plicity. Ideas such as SU 3 and current algebra were very exciting when they 
were introduced ; they supplied an appropriate terminology for the description 
of facts, they showed how certain phenomena are connected with others, 
but it does not appear that they wave brought us much nearer to an under- 
standing of the subnuclear world. The existence of three types of seemingly 
unconnected interactions is still an unsolved problem, although the close 
relation between electromagnetic and weak vector currents may be the first 



286 V. F. Weisskopf 

hint at some deeper connection. We also have no understanding yet of the 
nature of the electric charge with its double manifestation as electron and 
muon. 

High-energy physics today is an experimental science. We are exploring 
unknown modes of behaviour of matter under completely novel conditions. 
The field has all the excitement of new discoveries in a virgin land, full of 
hidden treasures, the hoped-for fundamental insights into the structure of 
matter. It will take some time before we can produce a rational map of 
that new land. 



REFERENCES 



[1] S. Adler: Phys. Rev., 143, 1144 (1966). 

[2] L. Van Hove: CERN publication 68-31 (1968). 

[3] D. Dorfan, J. Enstrom, D. Raymond, M. Schwartz, S. Wojcicki, D. H. Miller and 
M. Paciotti: Phys. Rev. Lett., 19, 987 (1967); S. Bennett, D. Nygren, H. Saal, 
J. Steinberger and J. Sunderland: Phys. Rev. Lett., 19, 993 (1967). 



Some Questions Concerning Adiabatic Transformations. 



G. C. Wick (*) 

CERN - Geneva 



1. - Introduction. 

The old questions connected with the so-called « adiabatic theorem » [1] 
still play an important role in many questions of quantum field theory such 
as renormalization, definition of the vacuum state, etc. It may therefore 
be of some interest to simplify or clarify the proof of certain results, even 
though they are generally regarded as well known. We refer in particular 
to the following: A very common situation arises when a coupling constant 
is switched on exponentially, g{t) = g exp [at] ; the adiabatic limit is then 
obtained, of course, for a-^0 + . For finite a, the state \p g attained by the 
system at, say, the time t = 0, satisfies a well-known differential equation 
with respect to the coupling constant g. This equation was obtained by 
Gell-Mann and Low [2] by means of a perturbation expansion in the inter- 
action representation. We note in Sect. 2 that a much simpler and more 
transparent derivation can be given, without resorting to a perturbation 
expansion. Our main topic, however, is the limit a -> 0. As is well known, 
the state vector \p g contains a phase factor exp [— id(g, a)] which becomes 
singular in the limit a^O, in the sense that d~l fa. Physically, this infi- 
nite phase must be the integral over time, from / = — oo to t = 0, of AE(g)fh, 
where AE(g) is the level shift produced by the perturbation; when a is finite, 
this shift operates effectively for a time of the order of 1/a, hence the above- 
mentioned result. Formally, the existence of the singularity can be easily 
checked [2], if one examines the behaviour, in the limit a -> 0, of the suc- 
cessive terms in the perturbation expansion; one finds that the term of 
order g n contains singular terms in a, of order or 1 , a -2 , ..., or 71 . It is intui- 
tively plausible, but not obvious mathematically that these terms arise from 



(*) Permanent address: Columbia University, New York. 



288 G. C. Wick 

the expansion of the exponential phase factor. If this is true, it becomes 
possible to eliminate the singularity at a = by renormalization of ip g , for 
example [2], by the definition 

(1) Xg = Wgf( ( P> W) » 

where (p is the initial value (at t = — oo) of the state vector. Here (..., ...) 
indicates as usual the scalar product of two vectors. It is, of course, a fun- 
damental result that % g should have a finite limit for a = 0, because it is 
this limit, which is then shown to be an eigenstate of the perturbed Hamil- 
tonian, as stated by the adiabatic theorem. 

Now Gell-Mann and Low content themselves with asserting that all the 
above statements can be verified « up to any desired order in g », i.e., pre- 
sumably as far as one's patience and willingness to be convinced allow. 
The direct calculations actually become quite laborious very soon. The 
question, however, is so fundamental and simple that it should be settled 
completely for an arbitrary order n. We show in the following that this is 
indeed possible by utilizing the differential equations in g satisfied by the 
state vectors ip g and % g , equations already given by the above-mentioned 
authors. The connection between the adiabatic solution and the Rayleigh- 
Schrodinger perturbation series is thus completely established. 



2. - The differential equations. 

The problem to be dealt with is the following [2]: The Hamiltonian of 
interest is of the form 



(2) 3? = ^(s) = =^o + s"i> i 
One asks for a solution of the time-dependent Schrodinger equation 

(3) / gy = {^o + S exp [oct] ^} y , 

namely, the solution which for t -> — oo behaves asymptotically according to 

(4) y)~<pexp[—ie t], 

where q> is a normalized eigenstate of the unperturbed Hamiltonian corre- 
sponding to a nondegenerate eigenvalue e : 

(5) (^ -e )<p = 0. 
In eq. (3), a is a real constant > 0. 



Some questions concerning adiabatic transformations 289 

We now denote by xp the value of this solution for / = 0. More ex- 
plicitly, let ip(t, g, a) be the unique solution of eqs. (3) and (4), then 

(6) xp g = yj(0, g, a) . 

We derive a differential equation with respect to g, satisfied by ip g , as fol- 
lows. Consider the function: 

(7) xp x {t) = exp [is r]ip(t + r, g, a) 

which, apart from a phase factor, is obtained from the solution ip by a time- 
displacement t. Clearly tp ± satisfies the asymptotic condition (4), and also 
the Schrodinger equation (3), for a different value of g: g 1 = gexp[ocr]. 
Therefore, ip x {t) = ip(t, gexp [ocr], a) or: 

(8) y(t + t, g, a) = exp [— ie r] ip(t, g exp [ar], a) . 

Differentiating with respect to r and setting r = 0, we can express the time 
derivative in eq. (3) in terms of txg(dtp/dg) — ie ip. At t = 0, eq. (3) becomes 

(9) iocg^- = {^(g)-e }y> g , 

which is Gell-Mann and Low's eq. (A. 5). 

This derivation does not depend on a perturbation expansion and it can 
be easily generalized. For example, assume 

(10) JTfe) = 3f + gMT x + £ 2 ^ 2 

(or even a polynomial in g of order n) and then consider the time-dependent 
Schrodinger equation in which J^(g) is replaced by J4?(gexp[oct]). Then 
everything goes through as before, and eq. (9) is obtained. In the following, 
however, we shall restrict ourselves to the simple case, eq. (2). The gen- 
eralization (10) is not without interest, however, since Lagrangians and 
Hamiltonians with quadratic terms in the coupling constants are quite 
common. 

Finally we notice with Gell-Mann and Low the following equations of 
which the second one is a differential equation for the renormalized state 

vector xg' 

9 

(11) iccg g- {(p, ip g ) = (<p, [ &, (g) — £ ] y> g ) = g(<p, 3^np g ) , 

(12) iccg-^ = {^(g)-e }x g -Xgiocg^ln((p, y> ) . 

19 



290 G. C. Wick 

Both equations, of course, follow from eq. (9). In eq. (12), furthermore, 
the last term can be modified as follows 

8 

(13) iocg g- In (<p, y> ) = g(<p, 2P X % g ) , 

so that finally 

(14) iocg ^ = {jT(g) - E(g, a)} Xg , 

(1 5) E(g, a) = e + g(q>, Jt x Xg) • 

As we shall see, in these two last equations, one can go to the limit a = 
without any trouble, and E(g, 0) is then the perturbed eigenvalue [of the 
operator Jf(g)]. 



3. - The adiabatic limit. 

It is customary to obtain the perturbation expansion of ip g 

(16) V9 = v+ ^^- n ^- 

l E Q — M + icc(n—l) e —Jf + ioc 

from the time-dependent equation (2). From this, in principle, one can ob- 
tain the expansion of % g , eq. (1), but, as we have said, this is rather cum- 
bersome. Notice, however, that (16) can be obtained directly from eq. (9). 
It should be almost equally simple to obtain an expansion for Xg directly 
from eqs. (14) and (15). For g = 0, Xa = V>g = <P1 assume therefore 



(17) Xg = Zg n <Pn(cc), 

where (p = <p. Notice also that, from (1): (99, Xa) = h so that we must nave 

(18) (q>,<Pn) = &w>\ (n = 0, 1,2, ...). 
Substituting (17) into (15) we obtain 

(19) E(g, a) = e + 5 g n e n (cc) , 

where 

(20) e»(a) = (<p, =3f i^n-iCa)) . 



Some questions concerning adiabatic transformations 291 

In particular, £ x (a) = (<p, Jf^) is the first-order perturbation of the energy 
level, and is independent of a. Next substitute (17) and (19) into (14) and 
obtain the recurrence relation 

(21) (s — JT + iccn)(p n (cc) = ^ifn-^oc) — (99, ^ X (p n - X {(£)) <p — - 

n-X 

~^£n-m(oc)(p m (oc) ; (n = 1, 2, ...). 

m=l 

Thus the nth term in the expansion of % g is at first sight much more com- 
plicated than the corresponding term in (16). It has, nevertheless, much 
simpler properties in the limit a = 0. Notice in fact that the solution of 
an equation (e — ^f + iocri)(p n = oj n or 9% = (e — Jf + zan) _1 a> w becomes 
singular as a->0 only if (99, a> n ) ^ 0. But the right-hand side of (21) may 
be rewritten 

n-l 
(21') co n = Aye x y n _ x — ^ e n -m(pm , 

where A is a projection operator (projection on the subspace orthogonal 
to 99). Together with (18) this implies that oj n is orthogonal to 99, and 
9% = lim (p n (oc) exists, and may be obtained by solving the recurrence equations 

ra-l 

(22) (fi — J5f 0)9% = A^ x (p n _ x — 2 £n-m<Pm , 

m=l 

(23) e n = (rp, ^ x y n _ x ) . 

These are, however, precisely the equations of the Rayleigh-Schrodinger 
expansion of the perturbed wave function and perturbed eigenvalue s J r AE(g). 
The cancellation of the singular phase factor is therefore demonstrated to 
arbitrary order. 



4. - Remarks and extensions. 

It is clearly possible to extend the above calculations in various direc- 
tions. For example the conclusions of the last Section could be easily extended 
to the case of eq. (10). Also, one could expand 9%(a) and e w (a) in powers 
of a and examine non-adiabatic effects; for example, terms up to order oc 
in e w (a) are easily connected to the normalization factor in the denominator 
in eq. (1), since one can prove, by means of eq. (13), that 

9 

(24) Info y>„) =- '- U E (g')^—if ^ #>+... , 

a J g n=l n 



292 G. C. Wick 

where 



e£> = (de w /da) a „ • 



We want to deal briefly with a different extension. The methods of perturba- 
tion theory, adiabatic switching on, and renormalization, have been applied 
not only to the Schrodinger equation but also directly to the field equations 
in the Heisenberg representation, as, for example, in the work of Kallen [3] 
and of Yang and Feldman [4]. The most important difference from the 
previous case is that the equations are nonlinear. Nevertheless a method 
similar to that of Sect. 2 leads to some interesting equations. Consider as 
an example the scalar field equation 

(25) (□ + m 2 )<p(x) = — X exp [xt]qP(x) . 

Here x = (x, t) and the coupling constant X is « switched on » by the ex- 
ponential factor. One seeks a solution which satisfies the boundary condition 

(26) <p(x)~ (p (x) when t^ — oo, 

where cp is a solution of the homogeneous equation (□ + m 2 )cp = 0. For 
simplicity, we limit ourselves to the nonquantized version of the theory [5]. 
We notice that instead of the boundary condition (26) we may consider 
more generally 
(26') (p(x) ~ <p (x + £) when * -» — oo, 

where I = (0, t) is a displacement in "the time direction. In fact <p (x + I) 
is also a solution of the homogeneous equation (and in the quantized case, 
it satisfies the same commutation law as <p (x)). Now if we call @(x; X, £) 
the solution of (25) and (26'), it is easy to prove by reasoning as before, 
that satisfies the functional equation: 

(27) &{x ;X,£) = 0(x+£;X exp [- or], 0) , 

in other words depends on the three variables t, X and r only as a function 
of two variables: t+r and Aexp[— at]. Hence 0(x; X, |) satisfies iden- 
tically the differential equation 

By differentiation one derives from this 

(29) ^F = -8^ + 2fla 8A8i + aA 8F + aA 8l' 



Some questions concerning adiabatic transformations 293 

We now write the differential equation (25) at time t = 0, replacing the 
second derivative with respect to time by means of (29). The boundary 
condition (26) is replaced by 

(30) = <p (x, t) when A = . 

The equation obtained may play a similar role as eq. (9). The solution can 
be expanded in powers of X, and the coefficients of X n (n = 1, 2, ...) obey 
a set of recurrent equations. We do not wish to deal with this any further 
here, but we shall content ourselves with the remark that the method seems 
to be of some interest also in the more complicated case of the quantized 
theory. 



REFERENCES 



[1] M. Born: Zeits. f. Phys., 40, 167 (1926); M. Born and V. Fock: Zeits. f. Phys., 51 

165 (1928). 
[2] M. Gell-Mann and F. Low: Phys. Rev., 84, 350 (1951). 

[3] G. Kallen: Helv. Phys. Acta, 22, 637 (1949); Ark. Fys., 2, 187 (1950); 2, 371 (1951). 
[4] C. N. Yang and D. Feldman: Phys. Rev., 79, 972 (1950). 
[5] The complications of the quantized case are discussed by: W. Zimmermann: Commun. 

Math. Phys., 6, 161 (1967); 10, 325 (1968). 



Range and Straggling of Muons. 
R. R. Wilson 

National Accelerator Laboratory - Batavia, III. 



The range of muons is calculated including radiation, pair production, 
and nuclear effects. It is shown that the large individual energy losses char- 
acteristic of these processes reduce the average range by a factor of In 2 
from that which one would get on the basis of simply integrating the aver- 
age energy loss. The fractional straggling first increases as radiation and 
pair production effects become important and then decreases as the energy 
is further increased. 



1. - Introduction. 

The study of the passage of fast-moving particles through matter has 
been important since the early days of nuclear physics [1]. Many of the 
experimental techniques of detection and measurement of particles depend 
on such specific properties of penetration as the total range or as the specific 
energy loss. Protons and pions at energies below a few hundred million 
electron volts have a well-defined average range, but the effects of nuclear 
collisions obscure this definite range at higher energies: The track of the 
incident proton becomes completely lost in the accompanying nuclear shower 
of secondary particles at energies higher than a billion electron volts. Indi- 
vidual electrons never show a well-defined range: At low energy, multiple 
scattering causes them to diffuse through matter; and at high energies a 
shower conceals the initial electron. 

Muons are more satisfactory particles to consider from this point of 
view because at low energies multiple scattering is not too serious, and at 
high energies the effect of nuclear collisions is small. On the other hand, 
at high energies, bremsstrahlung and direct pair production do occur to in- 
crease the energy loss above that due to ionization. The nature of the large 



Range and straggling of muons 295 

individual energy losses due to radiation markedly increase the fluctuations 
of individual track lengths. 

The present study is concerned with a quantitative evaluation of the range 
and straggling of muons at very high energies. Briefly stated, the range is 
smaller by a factor In 2 from what one would calculate by neglecting fluc- 
tuations; on the other hand, the fluctuations on the ranges of individual 
tracks are smaller than might be expected intuitively. 



2. - Range calculations. 

First let us calculate the range assuming only bremsstrahlung losses be- 
cause this can be done more or less rigorously, then we will include pair 
production, nuclear absorption, and ionization. The calculation will parallel 
that made by the author for electrons [2]. Bethe and Heitler [3] give the 
energy loss of a particle that has traversed a thickness of matter t. They 
approximate the radiation spectrum by 

(1) a{k)dk = dkdt 



E\n[EI(E—k)Y 

where a(k)dk is the probability of the muon energy E radiating a photon 
of energy k in passing through a distance dt measured in units of shower 
length, i.e., for this application in muon radiation lengths divided by In 2. 
Then they find that the probability of the particle of initial energy E having 
an energy E after traversing a finite distance t is 

(2) w(y,t) = (t-l,y)\/(t-l)\ 

in terms of the incomplete gamma-function (t— 1, y)l, where y = \n(EJE). 
From this one can find [4] the result we seek, namely 

(3) w(y, t)dt = e~ y ydt/t\ . 

For large values of y and t, the above equation can be approximated by 
the Gaussian form 

(4) w(y, t)dt = (2jiy)-l exp [- (t - yf/2y) dt . 
From this we see directly that the mean range r is 

(5) r = y max , 



296 R. R. Wilson 

where y max is the value of y at the peak of the Gaussian, and that the root- 
mean-square straggling of the range s is given by 

(6) s = jLx = r* • 

21. Ionization loss. - If only radiation contributed to the energy loss, 
the range and the straggling would be infinite, however, as the energy de- 
grades, ionization losses become important and allow us to evaluate y max - 
Expressing eq. (5) in terms of energy and then differentiating the mean range 
with respect to the initial energy gives the average radiation loss on travelling 
a distance dt, 

< 7 > -£-*■ 

That this is neither obvious nor trivial is clear if we remember that we are 
using shower units of length which introduce the factor In 2. 

Now let us express the energy in units equal to In 2 times the critical 
energy /S, i.e., the energy lost by a muon to ionization in going a distance 
of one muon radiation length. Here we are making the rough approxima- 
tion that the ionization loss is independent of energy: we will examine the 
validity of this later. We add the ionization loss to (7) to get 

(8) — dE/dt = E+l 

and integrating this over the energy gives the mean range, 

(9) r = log (£ + l)> 
which in units of radiation lengths becomes 

(10) r = ln21n[(£o//81n2) + l]. 

Now let us turn to straggling which is manifest in the distribution de- 
scribed by eq. (3). Equation (6) shows us rather surprisingly that on the 
basis of radiation loss alone the fractional straggling sir varies as 1/r*, i.e., 
becomes smaller as the energy increases. Actually the straggling at low 
energies will be less than given by (6) because the energy loss due to ioniza- 
tion has been neglected. The energy loss due to ionization for a muon trav- 
eling a distance r is just equal to r in the peculiar energy units of the above 
theory, hence a fraction of the range r/E can be ascribed to ionization loss. 
The straggling of this fraction will be less than 1 % ; hence for any energy 



Range and straggling of muons 



297 



for which this calculation can apply, we can neglect it completely. The re- 
maining fraction of energy that is lost to radiation processes (1 — r/E ) will 
vary roughly as given by eq. (4), which will only be valid when the fraction 
(l — r/E ) is large. As a rough interpolation formula, suggested by Monte 
Carlo calculations [2], we can write 



-('-if 



and remember that the Gaussian distribution has a cut-off at a range equal 
to E in shower units or E/fi in radiation lengths. 

2'2. - Pair production and other energy losses. - Up until now we have 
neglected direct pair production as well as nuclear interactions. In fact, 
the loss of energy due to the direct production of electron pairs is compa- 



10' 



1 1 — I I I 1 1 



I I I 1 1 



1 1 I 1 . 1 I I I. 




10 



100 10 3 

E (GeV) 



10< 



FIG.1 



Fig. 1. - The range and straggling in various materials. 



298 R. R. Wilson 

rable (within about 10%) to the loss due to bremsstrahlung [5]. The energy 
loss due to nuclear interactions amounts at most to a few percent of the 
total loss, being relatively larger for light elements and at high energies. 
On the basis of a Weissacker- Williams consideration of direct pair produc- 
tion, we can expect the atomic shielding factor to be almost identical to that 
for bremsstrahlung. Thus, except for the lowest energy losses where the 
mass of the created pair becomes significant, we can expect the two pro- 
cesses to be roughly the same. 

In order to include these effects, I suggest that we define a muon inter- 
action length in place of the muon radiation length that we have been using 
thus far so that the above theory will be valid. Until a more exact calcu- 
lation of the muon interaction length is made, I suggest that we use simply 
one half of the muon radiation length. On this basis, the range and strag- 
gling in various materials has been computed and is plotted in Fig. 1 . The 
range in uranium is significantly smaller than that given in ref. [5] where the 
effects of fluctuations were neglected. 



REFERENCES 



[1] H. Bethe: Ann. Phys., 5, 325 (1930); N. Bohr: Phil. Mag., 25, 10 (1913). 
[2] R. R. Wilson: Phys. Rev., 84, 100 (1951). 

[3] H. A. Bethe and W. Heitler: Proc. Reg. Soc. {London), 146, 94 (1934). 
[4] Using 

y 

\x l e- x dix = t\ — e-V^t\lp\yP 

o 

which can be derived by successive partial integrations (for integral i), one obtains 
eq. (3) by taking the difference between t and (r — 1). This procedure was shown to 
me by S. Pasternack: originally, I simply guessed the result [3]. 
[5] R. H. Thomas: UCID 10010, July 2, 1964. 



The Basic SU 3 Mixing: co 8 ^ w 1# 

A. ZlCHICHI 

Istituto di Fisica delUUniversita - Bologna 
Istituto Nazionale di Fisica Nucleare - Sezione di Bologna 



1. - Introduction. 

The purpose of the present paper is to review a basic problem of SU 3 
symmetry, the (co-(J>) mixing, and to describe the first direct measurement 
of the so-called (co-<j>) mixing angle. 

l'l. The origin. - The (co-(j>) mixing has played a basic role in the under- 
standing of hadron spectroscopy and therefore in the discovery of SU 3 [1, 2]. 
In fact, without guessing the SU 3 symmetry-breaking it would have been 
impossible to establish its existence. The regularity of SU 3 symmetry-breaking 
follows the observed regularity of the other symmetry breakings (such as: 
isospin by electromagnetic interactions, and C, P by weak interactions), 
and represents an aspect of particle physics that is even more spectacular 
than the existence of SU 3 . 

As is well known, SU 3 symmetry is not an exact symmetry law. When 
medium-strong interactions are switched on (their strength in supposed to 
be about one-tenth of the strong interaction strength), SU 3 symmetry is 
broken. By assuming that the symmetry-breaking term is a coherent super- 
position of an SU 3 singlet and the eighth component of an SU 3 octet, the 
celebrated unitary symmetry mass formula, first derived by Gell-Mann [1] 
for a unitary octet and then generalized by Okubo [3] to any unitary multiplet, 
is obtained. This Gell-Mann-Okubo mass formula (linear for fermions, 
quadratic for mesons (*)) was found to be in good agreement with all observed 



(*) A simple argument to justify the rule of linear expressions for fermions and of 
quadratic expressions for mesons is due to Feynman, and is based on the fact that the 
mass term in the Dirac and in the Klein-Gordon equations is linear and quadratic, re- 
spectively. 



300 A. Zichichi 

mass values for the pseudoscalar meson octet, the spin | + baryon octet and 
the spin f + baryon decuplet (*). Surprisingly enough, the Gell-Mann- 
Okubo mass formula failed to explain the observed vector meson masses. 
More precisely, the situation concerning the vector meson octet was the 
following: from the known mass values of the K* and of the p mesons, 
raic* = 888 MeV, m p = 750 MeV. the isoscalar vector meson was expected 
to be at m I=0 = 930 MeV. But the co-meson mass was known to be at 
m u = 780 MeV. It was indeed very disturbing to find that the regularity 
of the Gell-Mann-Okubo breaking failed so badly to explain the observed 
spectrum of the vector mesons. This was the sitation in 1962 when Sakurai [5], 
on the basis of the discovery of the 1=0 vector meson at 1020 MeV (the 
cj)-meson) by the Brookhaven-Syracuse Group [6], put forward the proposal 
that the cause of the failure of the Gell-Mann-Okubo mass formula was the 
fact that the co and <j> mesons were two particles with identical quantum 
numbers, as far as spin parity, isospin, and (7-parity are concerned. The 
only difference (if any) between co and <j> would be their SU 3 attribute. But 
SU 3 is only an approximate symmetry. Therefore it is reasonable to suppose 
that the 780 MeV co-meson and the 1020 MeV (j>-meson are coherent super- 
positions of two pure SU 3 states: a pure SU 3 singlet co 1? and the eighth com- 
ponent of a pure SU 3 octet co 8 . On this basis Sakurai [7] was able to construct 
a simple dynamical model of basic SU 3 symmetry-breaking that could account 
for the success of the Gell-Mann-Okubo mass formula. 

It should be clear at this point that it is more appropriate to speak in 
terms of (cog-cO]) mixing, rather than (co-^) mixing (**). 

The crucial point was then to check if this mixing really exists in nature. 
For a long time the only experimental information on the (co-cj>) mixing was 
obtained from the observed mass spectrum of the various vector mesons. 
But this is more a way of adjusting the Gell-Mann-Okubo mass formula, 
than a measurement of the (cog-co!) mixing. 

1*2. Attempts to measure the (cog-coj) mixing. - A way of measuring the 
(cog-coj) mixing was suggested by Sakurai [7], who pointed out that if the § 



(*) To be more precise, at that time the tenth member of the J p = f + decuplet, 
the QT baryon singlet, was still unobserved, but only theoretically postulated [4]. Nev- 
ertheless, the famous equal-spacing rule, i.e., the proportionality of the mass M to the 
hypercharge Y, M = M (l + aY), which is a straightforward consequence of the Gell-Mann- 
Okubo mass formula, was very well satisfied for all the other observed members of the 
baryon decuplet JV |(1235), Y*(1380), and Ef(1530). 

(**) An analogous situation occurs in weak interactions where the (K°-K°) mixing pro- 
duces the physically observed K° and K^ states. Notice that K° and K°, like co 8 and to l5 
remain unobserved as particle states. 



The basic SU 3 mixing: cogi^to! 301 

width TV-*!! is calculated from the known p width and compared with the 
partial width T^kk, the mixing angle is obtained through the relation 

-^ = cos* 2 d . 

1 <[)->.all 

This is because in the decay <]> -> KK the coj cannot contribute. In fact to x 
can only be coupled to a symmetric bilinear expression involving two pseudo- 
scalar K's, which consequently cannot be in a state with J =\. 

The value of 6 deduced in this way has been controversial because of 
the different results obtained for the branching ratios of the cj> decays 
(<}> -> 3n and (f> -> KK) in various laboratories [8] : these results led to values 
of d with one standard deviation limits ranging from 0° to 55°. Other ways 
of determining 6, as for instance those suggested by Glashow [9], have also 
not led to any positive result. The first significant « indirect » value for the 
(cog-coi) mixing angle was obtained by Massam and Zichichi [10] via a world 
analysis of the nucleon electromagnetic form factors. 

1'3. The direct way. - All these suggestions were put forward and the 
corresponding attempts were made because the direct clean way of checking 
the (cog-coi) mixing hypothesis seemed to be quite remote from the experi- 
mental area. As we shall see later, this « clean » and « direct » way is either 
via the study of the electromagnetic decay modes of the to and (J> mesons, i.e., 

Ico ->- y -> e + e _ , 
9 -> y ->- e + e , 



or via their production through (e + e _ ) colliding beam machines: 
(2) 



e + e~->y ->co 
e + e~->Y-xj> . 



The major difficulty connected with reactions (1) is that the to and § 
mesons decay via strong interactions, and their electromagnetic channels 
are expected to be depressed by a factor of the order of a 2 (where a is the 
fine structure constant): This means that the branching ratios 

(3) r " +e_ 



-^co-vail 

and 

(4) ^: 



302 A. Zichichi 

are expected to be of the order of 10 -4 . Moreover, there were also difficulties 
associated with the production processes of the co and (j>. 

The to production could go with a reasonable cross-section via the reaction 

(5) 7r-+p-^co+n, 

but, as the w-meson mass is enveloped in the large width of the p mass, a 
large background of p's was expected to be present; thus it was necessary 
to choose, if possible, those experimental conditions where the p contribution 
is depressed in favor of the co production. 

The (J) production was observed with a reasonable cross-section in reactions 
where strange particles were present, such as K _ p-^A°(j), but was expected 
to be depressed by a large factor (due to the A-quark spin conservation), 
and in fact remained unobserved for a long time, in simple reactions such 
as [11] 

(6) 7r-+p^(j)+n. 

Finally, in order to observe w and § decays into (e + e~), it was obviously 
necessary to devise a large acceptance experimental apparatus, able to select 
and measure angles of emission and energies of the final products of reac- 
tions (5) and (6), which are neutrons and (e + e~) pairs. The feasibility of 
such an experimental program was shown in an unpublished paper by Dal- 
piaz et al. [12]. 

The difficulty connected with reactions (2) lay in the fact that (e+e~) 
storage rings had to be built. On the other hand, no serious problems of 
particle identification or of strong background could be foreseen in the 
study of these processes, the feasibility of the experiment being confined to 
the problems connected with the construction of moderate-energy (e + e~) 
storage rings [13]. 

As we shall see later, the first measurement of the (cog-coj) mixing was 
obtained via the study of reactions (1). 

As mentioned before, the (cog-o^) mixing was at first described by Saku- 
rai [5] using a unique mixing angle 6, but by now there are four (cog-coj) 
mixing angles quoted in the literature [5,6,8, 14-18]; the original 6, then 
6 Y and 6 N , and finally the generalized mixing angle d G . 

2. - The four mixing angles. 

2'1. - Schematic derivation. - We shall now try to review the origin of 
all these mixing angles. The starting point is: two particles with identical 
quantum numbers (J pa , I, Y) such as co 8 and (Oj will convert into each other 

(7) cog^co!, 



The basic SU S mixing: (Og^o^ 303 

because process (7) does not violate any conservation law but that of SU 3 
symmetry, which is broken by the moderately strong interactions. As it is 
impossible to switch off these interactions, process (7) will go. 

When two particle states can convert into each other, as in process (7), 
the inverse propagator that describes the mixed system can be shown to 
have the familiar form 

(8) D = AK 2 + BM 2 , 

where A and 5 are 2x2 matrices (if we want to describe mixing between 
two particles only), K is the quadrimomentum, and M is the mass of the 
two states. Let D , A , and B be the quantities defined above before the 
mixing starts. Without mixing the two matrices, A Q and B are diagonal, i.e., 





(1 Ov 




ft 


0> 


4> = 


■ 


B = 








\o 1/ 




\o 


ft> 



and the inverse propagator 

(9) D = A K 2 + B M 2 

gives the two propagators of the two unmixed states, each having momentum 
K and masses ftM 2 and ftM 2 (ft and ft are just numerical coefficients). 
The effect of mixing can be of two types. These two ways of treating the 
mixing between two particles have been discussed first by Coleman and 
Schnitzer [15] (CS) and later by Kroll, Lee and Zumino [16] (KLZ), who 
particularly emphasized the need of having two mixing angles. 



2' 1.1. Mass mixing. Here it is supposed that the effect of mixing 
(process (7)) is that of destroying the diagonality of the matrix B Q , which 
becomes B = B + 8B, without disturbing the matrix A . 

The problem is to diagonalize B and hence D without destroying the di- 
agonality of A Q . It is well known that in order to achieve this, the matrix 
that is needed can be an orthormal matrix. As the elements of a 2 x 2 matrix 
are four, and the orthonormality conditions are three, all the mixing can be 
described using a single parameter: the mixing angle 6, which is the angle 
first introduced by Sakurai [5]. The left side of Fig. 1 shows a synthesis 
of the above chain of arguments. 



304 



A. Zichichi 



(tog-o^) MIXING 



D = AK 2 + BM 2 



Without mixing: A = 



1 
1/' 



h 
ft 



Effect of mixing 


i) Mass-mixing 


ii) Current mixing 


\ 


1 


Destroy diagonality of 


Destroy diagonality of 


Bo 


A 


1 


I 


Problem: diagonalize 


Problem: diagonalize 


B = B + 8B 


A = A + 8A 


Without destroying diagonality of 






Without destroying diagonality of 


*-(i J) *" 


compare 


*-(!' 3 


\° V 






I 


i 


Matrix needed: 






Matrix needed: 



a P\ with orthonormal- 
y fi] ' ity conditions: 



y2 _|_ ^2 = 1 

ay + j5<5 = 



4 Parameters - 3 Conditions 

I 
Only one parameter 



a 

y 8 



no orthonormality conditions 



4 parameters: 

(2 coupling constants: # r ; £y 
2 angles: r , 0^ 

Correlated 

I 

— tan r = -^ tan 0^ = tan G 
Wa m, A 



Fig. 1. - (cog-Qj) mixing: schematic comparison between the « mass-mixing » type 
and the « current-mixing » type models. 



The basic SU 3 mixing: cog^o^ 305 

21.2. Current mixing. Here it is assumed that the effect of mixing 
is to destroy the diagonality of A , leaving B diagonal. The problem is now 
to diagonalize A = A + $A, without destroying the diagonality of B . 
Notice that B is diagonal but (unlike A ) not unit matrix. In order to diago- 
nalize A without destroying the diagonality of the nonunit£ , a 2 x2 matrix, 
without orthonormality conditions, is required. The mixing must threfore be 
described using four parameters, which can be expressed in terms of two coup- 
ling constants g Y and g N , and of two mixing angles 6 Y and 6 N (here we use 
the same notation as KLZ). However, because of T invariance, A and B Q 
are symmetric matrices; this gives one condition for the four free parame- 
ters. This condition can be used in order to establish a relation between the 
two mixing angles Y and 6 N , i.e., 

tan 6 Y _ ml 
tan N ~ ml ' 

first derived by KLZ. This relation can be rewritten as 

(10) — tan0 F = -^ tanfljy = tan(9 G , 

thus allowing the mixing to be expressed in terms of the « generalized » 
mixing angle 6 G [18]. The right-hand side of Fig. 1 illustrates the above 
chain of arguments. 

2*2. Why are there all these complications! - After the introduction of 
the (wg-coi) mixing hypothesis by Sakurai [5], CS [15] emphasized that the 
Sakurai-type of mixing, called by them « particle mixing » and by KLZ [16] 
« mass-mixing », was not adequate enough to describe mixing between « vector 
particles ». This is because vector particles are believed to be associated 
with conserved quantities, and « mass mixing » is incompatible with this 
requirement, as can be easily shown with the following example. Suppose 
that the inverse propagator D describes the isoscalar form factor of the 
nucleon (*) and that we choose the « mass mixing » model. After mixing, 
D becomes D, 

D = A K 2 + B M 2 ^^ D = A K* + (B + Sfi)M 2 . 

At K 2 = 0, D ^ D. But the value of D and D at K 2 = is related to 
the nuclear isoscalar electric charge (i.e., electric charge of the proton divided 



(*) Pole dominance is of course assumed. 

20 



306 A. Zichichi 

by two). The effect of « mass mixing » is to change the value of the nuclear 
isoscalar electric charge and this is unacceptable. 
If we choose « current mixing » we have 

D = A Q K 2 + B M 2 ^ oixias > D = (A + SA)K 2 + B M 2 , 

and at K 2 = it is D = D. This is the reason why « current mixing » is 
believed to be more adequate for the description of the mixing between 
vector particles. 

It is interesting to notice [15] that if the force mixing the particles is truly 
weak, « mass mixing » and « current mixing » are indistinguishable (as in 
the case of the (K°-K°) mixing which produces the physically observed states 
K° L and K°; in fact here the transition K°^±K° is a second-order weak 
interaction (*)). Notice that in the above models of mixing it has always 
been assumed that the mixing alters only the propagators and not the vertex 
functions. Notice also that both « current mixing » and « mass mixing » 
are compatible with the transversality conditions for the source of the vector 
mesons [16], i.e., the currents to which they are coupled are conserved 
currents. 



2*3. The crucial point. - The conclusion of all the above arguments is 
that the physically observed states <o and <j> are mixtures with certain per- 
centages (%) of two pure SU 3 states, co 8 and o^: 

CO = (%)G>8 +(%)*>!, 
<|> =(%)** +(%)<■>!• 

The problem is how to measure these percentages. 

Suppose we have a selective interaction, i.e., an interaction which is 
coupled to co 8 and not to o^. If we can find such an interaction, we can then 
see how much co 8 there is in the physical states co and ej>. 

There is a good candidate for this selective interaction: the electromag- 
netic interaction. In fact, remember that all known particles obey the 
famous Gell-Mann-Nishijima relation 

(11) g = / 3 + - + nothing. 



(*) A simple way of seeing why in this case there is no difference between mass mixing 
and current mixing is to notice that CPT implies w K » = m s „, and therefore B turns out 
to be a diagonal and unit matrix. 



The basic SU 3 mixing: cog^coj 307 

It could be argued that even if the electric charge Q of all elementary par- 
ticles has no contribution from quantum numbers which are not I 3 and Y, 
the electromagnetic current J^ can still contain a singlet SU 3 term: /^ = 
= /£ 8) + J^- I n fact, if the fourth component of Jft has vanishing volume 
integral, then: 

(12) Q =J4 8 Kx, t)d*x +jji 1 Kx, O&x , 



and the Gell-Mann-Nishijima relation (11) remains unaltered. Here the 
difficulty with the electromagnetic current becomes clear. In fact the octet 
part of the electromagnetic current is a U-spin singlet (U transformations 
leave the electric charge invariant). All SU 3 predictions based on £/-spin con- 
servation alone cannot distinguish between the octet part and the singlet 
part of the electromagnetic current. In order to measure the octet part Jj?> 
and the possible existence of a singlet part J™ in the electromagnetic current, 
it is necessary to devise an experiment where these two parts can be directly 
observed ; the cleanest known example is the measurement of the (e + e~) decay 
rates of p, o>, §. In fact, if we believe in the one-photon approximation, 
these decays go via the following Feynman diagrams: 




The octet part of the electromagnetic current J (8) couples to the isospin (i.e., 
the p meson) and to the hypercharge Y (i.e., the <o 8 part of the co and § 
mesons). The singlet part of the electromagnetic current J^ ] couples to the 
(Oj part of the co and § mesons. Therefore if we measure the decay rates of p, 
oj, and <j> into (e+e~), we do study the coupling of the photon to the isospin, the 
hypercharge, and the SU 3 singlet. 



308 A. Zichichi 

To recapitulate, we have said that we wanted a selective interaction, 
coupled only to co 8 , in order to check the (ov^i) mixing hypothesis; but 
what we have found is a « good candidate » for the required selective inter- 
action. 

2'4. Conclusion. - The study of co -> e + e~ and <J> -> e+e~ will allow the 
(cog-co^ mixing hypothesis (in the limit where the photon is not coupled to Wj) 
to be checked; the comparison of the decay widths .Tw-^e+e-, i^-^+e- with 
.Tp-Hs+e-, will allow checking of the consistency of the selectivity hypothesis 
we made on the electromagnetic current, i.e., the SU 3 nature of the electro- 
magnetic current. 



3. - Vector-meson photon interaction. 

3' 1 . The first derivation. - In order to study the processes 



(13) 



p -^e+e" 
(o -> e+e~ 
i -> e + e~ 



it is necessary to know how to describe the coupling of a vector meson with 
the photon ; in fact, as mentioned above, these processes, in the one-photon 
approximation, are represented by the following Feynman diagram, where 
V stands for the three vector mesons p, co, cj>. 




Gell-Mann and Zachariasen [19] were the first to treat the problem of 
vector-meson photon interaction and to find out the effective vector-meson- 
photon coupling constant on the basis of vector-meson dominance. Their 
argument is as follows. Consider the 7r electromagnetic form factor (EMFF), 
F n (q 2 ). If we assume p-meson dominance, i.e., that the isovector photon is 
always coupled to the p-meson, then the pion EMFF will be given by 
the following expression: 



(14) F n {q 2 ) - 



q 2 ' 



The basic SU Z mixing: cog^coj 



309 



which can be easily derived by inspecting the corresponding Feynman dia- 
gram for the elastic (e-7t) scattering, where g Pr is the effective (p-y) coupling 





constant and g pmz is the (p-n) coupling constants and {m*-\-q 2 )~ 1 is the p 
propagator. At q 2 = 0, the pion EMFF is by definition equal to 1 (in units 
of the electron charge): 



therefore : 



(15) 



FJjS)=l 



Spy 



Spy SpTCTZ 



mi 



m 



gprnz 



If we introduce the universality condition for the p-meson hadron coupling, 
gffim=f P 'Ig i Spy becomes: 



(16) 



_m\ 

Spy f 
Jp 



where f p is the coupling of the p-meson to the isospin current. In fact uni- 
versality of the p coupling to the hadrons means that the coupling of the 
p-meson with its source density, the isospin current, is universal (at q 2 = 0), 
i.e., the ratio of amplitudes for any hadronic state A going into (A+p) and 
any hadronic state B going into (B+p) is just proportional to the ratio of the 
appropriate /th components of the isospin of A and B: 



A^±A+p 
B^±B + p 



If 
I? 



3' 2. The question of gauge invar iance and pole dominance. - The simple 
relation (16) was obtained by Gell-Mann and Zachariasen [19] by treating 
the coupling between the vector mesons V^ and the electromagnetic field A^ 
in the simple way 



(17) 



e'Vp-Ap , 



310 A. Zichichi 

where « e » is the electromagnetic coupling constant. This interaction pro- 
duces pole dominance but manifestly violates gauge invariance. 

Feldman and Mathews [20] remarked that in order to have a gauge in- 
variant electromagnetic interaction between the vector mesons and the electro- 
magnetic field, it is necessary to work with interaction terms of the type 

(18) G^-F^, 

where G^ v is analogous to 

F — ^ Av ^ A/x 

» v ~~ ax 8x ' 



for the vector mesons, i.e. 



av 1 _dv. 



The interaction (18) has the advantage of being obviously gauge invariant. 
It is in fact constructed using the fields instead of the potentials, but gives 
no pole dominance. In fact (*) 

/ 1Q \ r • f _ ** *" • rfi 

\ Ly ) "a" r i xv ~~~ Q z i m 2 v ' 

and this expression vanishes for q 2 = (real photons) (**). 

At this point it seemed there was no way out: one either had to 

i) choose e-V^-A^, then get pole dominance but lose gauge invariance; 

ii) or choose G^-F^, then keep gauge invariance but lose pole domi- 
nance in the sense that the interaction of vector mesons with real 
photons vanishes. 

3'3. The Kroll-Lee-Zumino theory. - The solution to this trouble was 
found by Kroll, Lee, and Zumino [16], who were able to reconcile pole 
dominance and gauge invariance in the description of vector-meson photon 
interaction. For simplicity we will again consider only the p-meson photon 
term. Their argument goes as follows: Add to formula (19) another term 

K-e-Jfi, 



(*) For simplicity we include only the p-meson photon term. The w and § terms 
are analogous but are longer because of their mixing. 

(**) The interested reader can find these points discussed further in: The Nature of 
the Photon, lectures given by A. Zichichi at the Enrico Fermi International School of 
Physics, Varenna, September 1969. 



The basic SU 3 mixing: cog^coj 311 

where I is a constant to be chosen later. Thus the total interaction 
term is 

(20) k{b„ • yp) + -^rh <f = -¥rK ^ 2 + m p) + ^ • 

J ^ ** q 2 + rm q l + m* H 

In the KLZ theory, K turns out to be: K = — \, as a consequence of the 
physical requirement that the expression (20) should vanish for # 2 ->oo. 
The interaction term (20) then becomes: 



(21) 



?2 _l_ M/.2 



+ m i 

Pole dominance is re-established in a gauge invariant theory of vector-meson 
photon interaction. It should, in fact, be emphasized that the « pole domi- 
nance » result (21) is achieved by KLZ not via the introduction of a term 
of the form q^A^ in the Lagrangian, but by establishing a special relation 
between the « direct » source term Jfi'A^ and the gauge invariant p-meson 
photon interaction G^F^. For clarity we draw the Feynman diagrams cor- 
responding to these terms: 




direct photon-hadron coupling 
(only isovector hadronic current) 

gauge invariant 

p-meson photon interaction 



A JP 



G F 

[IV flV 



An interesting point to remark is that as far as the effective coupling be- 
tween vector mesons and real photons is concerned, the KLZ result coin- 
cides with that of Gell-Mann and Zachariasen: 



= .K 

Spy e i- » 
Jp 

and analogously for co and <}>. 

In the language of KLZ, the interaction of the electromagnetic field with 
p, co, § takes place through the coupling of the electromagnetic field with the 
isospin current J* and the hypercharge current J*, whose explicit expres- 



312 A. Zichichi 

sions are (*) 

(22) 



I [AqI 



(23) J8 = ( cos r . /n 2.^_ sin0r . /M 2. o> ^ j 

JY 

where q^, eo^, <f> are the p, <o, cj> field operators, and r is the hypercharge 
mixing angle already mentioned. Notice that the electromagnetic field couples 
only to J% and J^ (the exact validity of this statement has already been dis- 
cussed in Sect. 2'3), and from formulas (22) and (23) the effective (p, co, (j>)-y 
coupling constants are derived to be: 



(15a) 



#PY e ' f ' 

J P 



P 

e--^sin0i 

JY 

e--7^cos0 ] 

JY 



The predictions for the (e + e~) decay rates of the vector mesons are, accord- 
ing to KLZ [16] (and to previous estimate [19, 21, 22]): 

(24) T^e- = J- (jr 2 j m p , 

(25) /^-ve+e- = °j l-^W-sin 2 ^ , 

(26) ^^e+e- = j I-72) m^ cos 2 B Y , 

(for the notation we follow KLZ). 

In these expressions SU 3 symmetry has not been used. If SU 3 is valid, 
f Y = \/3f p and 6 Y = 0. If we assume the « naive » SU 3 symmetry-breaking, 
then 6 Y = 35° and we obtain the well-known relation between the partial 
decay widths of the various vector mesons: 

(27) -^p->e+e- .' -^co->e + e- ' ^jj-^+e- = 9 .' 1 ! 2w , 



(*) Notice that J* and /® are th third and eighth components of /^ 8> . However, the 
SU 3 relation between / and / will be fixed later. 



The basic SU 3 mixing: cog^o)-,^ 313 

where w is an unknown factor resulting from the fact that the mass of the 
§ is different from that of the p and co. 

3'4. The generalized Weinberg spectral function sum rules. - For the pur- 
pose of checking the (o) 8 -«i) mixing hypothesis, it is sufficient merely to show 
that (o -» e+e~ and <j> -> e+e - both exist (*). Further developments of the 
mixing theory can be checked if we compare the partial decay rates (25) 
and (26), which give us 



tanfl, - V "V r « 



'Y 



and in terms of the generalized mixing angle 6 (see eq. (10)) : 



(28) tan0 o = = — tan d Y . 

Vffl^ • /*_*+«,- ™* 

As mentioned before, in order to have a self-consistency check on the 
SU 3 nature of the photon, it is necessary to compare iV^+e- an d -JV>e + e- 
with Tp^+e-, using for example the relation (27) (where it should be empha- 
sized that the factor « w » remains unknown). 

A more stringent relation can be established between -T p _>e + e-> -To-ws+e- 
and TV^+e- if we believe in the First Generalized Weinberg Spectral Function 
Sum Rule (FGWSR). Great interest in the Weinberg Sum Rules [23] was 
sparked from 

m Ai = V2m p , 

obtained from the first and second WSR, assuming pole dominance and the 
KSFR relation [24]. The two Weinberg Sum Rules related objects carrying 
the same isospin. The generalization produced sum rules relating objects of 
different isospin. According to Das, Mathur, and Okubo [17], and to Oakes 
and Sakurai [18] (we shall refer to them as DMS and OS, respectively) the 
generalization of the first WSR establishes the following relation among 
the (e+e~) vector meson decay rates: (**) 

(29) $ m p ■ Tp^+e- = ra w • r a ^ +e - + m$ • r^_^ +e - . 



(*) Assuming the selectivity of the electromagnetic interaction (see Sect. 2" 3 and 2'4) 
(**) This relation can easily be derived using eqs. (31), (15a), (24), (25) and (26), and 
remembering that the isovector coupling of the photons is -y/3 times stronger than the 
isoscalar coupling. 



314 A. Zichichi 

Furthermore, the « current mixing » result [16] between the two mixing 
angles B Y and 6 N (see relation (10)) is also derived from the 1st GWSR [17, 18] 



(30) tan0 G = ^ m « r <*^- = ^ tan0F = ^ tan ^ 

This seemed to imply [18] that in the vector-meson dominance approximation, 
the « current mixing » model of KLZ and CS is the only theory of (co 8 - w i) 
mixing which is compatible with the fst GWSR. As shown later by Majum- 
dar [25], Weinberg's first Sum Rule and the vector dominance hypothesis 
do not exclude any of the (cog-o^) mixing models, i.e., either « mass mixing » 
or « current mixing ». What happens is that a particular model of the 
(cog-o^) mixing demands a particular form of the spectral function sum rule. 
For example, in order to have relation (30), it is necessary to assume the 
Schwinger term between J* and J { * ] to be zero [25], besides assuming the 
1st GWSR and the vector dominance hypothesis. 

Relation (29) is more stringent than (27) ; however, it gives no predictions 
for 6 Y . In order to predict 6 Y , a precise model for SU 3 symmetry-breaking 
is needed. Various models have been presented in the literature: 

i) the quark model of Van Royen and Weisskopf [26] ; 

ii) two models of the « mass mixing » type by KLZ [16]; 

iii) three models of the « current mixing » type by KLZ [16], DMO [17], 
and OS [18], respectively. 

It turns out that the two models of KLZ and OS are in fact identical [27]. 
We shall just mention one point of the DMO and KLZ + OS models in con- 
nection with the Generalized Weinberg Sum Rules. 

In fact, as pointed out by Das-Mathur-Okubo [17] and by Sakurai [28], 
the 2nd GWSR must be abandoned as long as vector meson dominance 
approximation is considered valid. This is because, if we assume that the 
spectral functions are dominated by the know vector mesons p, co and §, 
we have 

(31) from the 1st GWSR: g -^ = ^ + ^f , 
y m$ m& m% 

(23) from the 2nd GWSR: gg Y = gl r + g| Y . 

For the consistency of eqs. (31) and (32) it is necessary that m\ = ml> =w|, 
which does not agree with observation. Das, Mathur and Okubo [17] 
and Oakes and Sakurai [18], proposed to change the 2nd GWSR a la Gell- 



Mann and Okubo, i.e. 



The basic SU Z mixing: cog^o^ 315 



DMO I dm 2 { Q 3 (m 2 ) + 3&(m 2 ) — 4p 4 (m 2 ) } 



r I 

2nd GWSR: {e 3 (m 2 ) — &(m 2 )}dm 2 changed into 



OsJ^{e 4 (iii a )+3 ft (iii 8 )-4 e4 (w a )}, 



where Q t are the spectral functions and the V s refer to the SU 3 component 
in the octet. It is interesting to notice that the DMO proposal [17] clearly 
implies that the Weinberg spectral function integral satisfies an octet-breaking 
formula, while the OS proposal [18] corresponds to the fact that it is the 
inverse propagator matrix for the current that satisfies an octet-breaking 
formula (*). These two proposals give different values for B Y . All the above- 
mentioned theoretical prediction will be reported in Fig. 15, Sect. 5, where 
they are compared with experimental data. 



4. - The first experimental measurement of the (cog-ooj.) mixing. 

4T. Introduction. - As mentioned in Sect. 1'3, there were two ways of 
attempting a direct check of the (cog-o^) mixing hypothesis : i) either by using 
strong interactions for the production of the vector mesons co and cj>, and 
subsequently detecting their rare decay modes into e + e~: 



(33) 



7r - +p->to+n 

'-> e + e~, 
7i-+p^(i>+n 

l-> e+e~, 



or ii) by using the electromagnetic production processes of co and rj> from (e+e~) 
collisions, and detecting the o> and <j) via their strong decay modes: 



(34) 



e + +e~->oo -> strong decay modes , 
e + +e _ ->(j> -> strong decay modes , 



(*) As emphasized by DMO [17], the 2nd GWSR is obtained assuming for the 
spectral functions p f 's, superconvergent conditions much stronger than those needed to 
obtain the 1st GWSR. This is why one expects the 1st GWSR to be much better than 
the 2nd GWSR, and therefore one tries to improve the last one. 



316 



A. Zichichi 



As reported by Ting at the Vienna Conference [29], the first successful 
experiment on the direct determination of the (ov^i) mixing angle was done 
at CERN by the Bologna-CERN Collaboration [30, 31] using reactions (33). 

It is obvious from the examination of the final states in reactions (33) 
that in order to perform the experiment it is necessary to have a large « neu- 
tron » detector and a large « electron » detector so as to be able to measure 
with good acceptance all particles present in the final states of the above- 
mentioned reactions. 

4*2. The experimental set-up. - A schematic diagram of the experimental 
set-up is shown in Fig. 2. It consists of the following: 

i) A system of « beam-defining counters » CUSR : C is a gas Cerenkov 
counter to anticoincide the electrons present in the primary beam; U is an 
important plastic scintillator counter used in the timing of the neutron; 
S is a very thin (0.05 cm) plastic scintillator counter in order to reduce as 
much as possible the interactions outside the H 2 target; R is an anticoinci- 
dence counter to remove beam halo. 

ii) A 40 cm long, 5 cm diameter H 2 target. A veto counter, not shown 
in Fig. 2, is placed behind the target in order to anticoincide noninteracting 
pions. 



ShMSng Weill 




team Count*™ 
not to Seal* 



Fig. 2. - Showing a schematic diagram of the experimental set-up. 



iii) Two electron detectors called « top » and « bottom ». In front of 
them there are coincidence counters and thin-plate spark chambers, which, 
for the sake of clarity, have all been omitted in Fig. 2. 

iv) Two neutron detectors, called « left » and « right », with anti- 
coincidente counters G T and G R in front of them to reject charged particles 



The basic SU 3 mixing: (Ogi^o^ 



317 



impinging in the « neutron counters ». These two identical neutron detectors 
had a sensitive surface and volume equal to 2.16 m 2 and 0.78 w 3 , respectively. 
A neutron detector is made of 12 elements of plastic scintillator, each 
having dimensions (100 X 18 x 18) cm 3 . Eeach element is viewed by two 
XP-1040 photomultipliers placed on its two small faces (see Fig. 3). The 



NDE 




H3^ 




Fig. 3. - a) Neutron counter assembly, b) The connections between a light guide 
and the photomultiplier base are shown. 



large volume of scintillator, in the particular geometrical arrangement chosen, 
allows a mean detection efficiency of about 26% in the range (40-^560)MeV 
neutron kinetic energy, for a laboratory solid angle of 0.14 sr at 4 m radial 
distance from the centre of the H 2 target. An interesting feature of this 
instrument is the accuracy achieved in locating incident particles; this 
accuracy is ±1.4 cm for charged particles, and ±2.5 cm for neutrons. The 
accuracies achieved for the time-of-fiigt measurement are ±0.35 ns for 
charged particles and ±0.7 ns for neutrons. It is interesting to note that 
the relative timing of all photomultipliers in the neutron counters could be 
equalized to ±0.1 ns. An example of this time-equalization is shown in 
Fig. 4, where t x is the time difference between the (£/) signal and the signal 
from any photomultiplier at one side of the neutron counter, t 2 corresponds 
to signals from the other side of the neutron counter, and is the difference 
between the two, obtained electronically (*). 



(*) For more details on this instrument we refer the reader to Bollini et al. [32]. 



318 A. Zichichi 



nsec 
+ 0.1 



I 














„ 














• 


• 




• 








• 


• — 


• 




• 


• 


e 


- 


• 




— # 

• 


• 






- • • • . 


• — 


• 




• 


• 


t 2 


• • — 

• 
• 

1 1 ■■ . 


• 
• 

1 1 


• 
JL . 


• 
• — 


— • 

• 
1 1 


• 


t, 



- 
-o.i - 



+ 0.1 -- 



- 
0.1 - 



+ 0.1 - 
- 



-0.1 - 



1 2 3 A 5 6 7 8 9 10 11 12 Right 

Number of the Counter 

Fig. 4. - Relative timing of the 12 elements of the neutron detector « Right ». 
The abscissa indicates the identification number of an element, the ordinate 

the relative timing. 



Nr/CH 
250 

200 

150 

100 

50 



POSITION 1.25 10 20 30 40 50 60 70 80 90 98 75 cm 




CHANNELS 



Fig. 5. - Spatial resolution of a neutron counter, as measured with a muon beam. 
Each of the peaks in this spectrum corresponds to a given position of the beam- 
defining telescope along the neutron counter. 



The basic SU Z mixing: cog^co-t 



319 



Typical data on position resolution and linearity of the neutron counters 
are shown in Figs. 5 and 6, respectively. In Fig. 5 the curves are labeled with 
the distance from the edge of the counter, and the spatial resolution for all 
positions in the counter is ±1.4 cm for charged particles. In Fig. 6 the ordi- 




200 Channels 



Fig. 6. - Position calibration of the neutron counters, showing the position of 
the muon beam as a function of the channel numbers. The ordinate is also 
labeled with the neutron scattering angle, corresponding to 4 m distance for 

the counters. 



nate is the distance from one edge of the counter, and the abscissa is the 
channel number in which the peak corresponding to a certain position (as 
shown in Fig. 5) falls. The counter is seen to be linear. 

Notice that there is a total of 24 elements. For all of them the above 
calibrations were repeated periodically in order to check the correct per- 
formance of the apparatus. For example, the neutron counter stability over 
a week is shown in Fig. la, where the time variation for 6, t x , and t 2 signals 
is plotted for each element of the neutron counter. The time stability of the 
neutron detector is remarkably good. Figure lb) shows the high-voltage 
variation over a period of one week for the « neutron-right ». 

The neutron detectors measure the times-of-flight t n and angles of emission 
6 n of neutrons in reactions (33), thus allowing a determination of the missing 
masses in reactions (33), i.e., of the mass of the produced meson. The mass 
resolution obtained with the above space and time resolutions depends on 
the kinematical region in the plane (t n , On) (see Fig. 8). It is ±4 MeV in the 
r-mass region, ±10 MeV in the co-mass region, and ±15 MeV in the cj>-mass 
region. 



320 



A. Zichichi 



o) 



So 








e 8 








d 


e 


t, 


*» 


I 






<9 








v> 












m 












Ui 












»- 












Z 














^4 


- 




- 










^ 
















o 
















m 
















5^ 












■ 






3 
















Z 


1 















0.2 



0.4 nt«c 



02 



0.4 rmc 



0.2 



0.4 



b) 



VOLT 



TYPICAL MAXIMUM VARIATION OF H.V. SUPPLIES OVER A PERIOD OF 1 WEEK 
1 VOLT PRODUCES A TIME SHIFT OF 15pMC 



RIGHT SIDE 1 



RIGHT SIDE 2 



PHOTOMULTIPIIERS 



Fig. 7. - a) Neutron counter stability over a week: the time variation for 6, t 1 , 
and t 2 signals is plotted for each counter, b) Typical maximum variation of 
the high-voltage supplies over a period of one week, for all the photomultipliers 
of one neutron detector. Notice that in fact a variation of 1 V produces a 15 ps 
shift in the time definition. 



The basic SU 3 mixing: cog^o^ 



321 



lT- + p— n+V° 
p„=2.12GeV/ c 




- 1 porticUs 



(M.V) 



20 



30 



-40 

-60 

MO 

150 
-250 



0« 



100 



200 



300 



400 



50° 



On 



Fig. 8. - The neutron time of flight t n over a 4 m path is plotted vs. the neu- 
tron emission angle n in the laboratory system. The ordinate on the right refers 
to the neutron kinetic energy. The kinematic curves are labeled by the corre- 
sponding neutron missing masses. The dashed lines indicate constant values 

of cos 0* . 



The electron detectors are shown in detail in Fig. 9. Each electron detector 
consists of nine elements, each one being made of a piece of lead followed 
by a two-gap spark chamber and a plastic scintillation counter. The first 
layer of lead is two radiation lengths thick ; the other layers are one radiation 
length thick. The over-all detector thickness is half a meter. Before the 
first lead layer there are two thin-plate spark chambers to allow precise kine- 
matical reconstruction of the events. Thus a long H 2 target could be used, 
when looking for rare events, without losing accuracy in the missing-mass 
measurement by the neutrons. The two detectors may be rotated independ- 
ently about a horizontal and vertical axis through the H 2 target. 



21 



322 



A. Zichichi 



LEAD 



R S 



BEAM 



H? TARGET 



T i ELECTRON DETECTOR 



LEAD 



LEFT 




RIGHT 

To neutron detectors 

(out of the plane of the page) 



ELECTRON DETECTOR 
'BOTTOM' 



Fig. 9. - Side view of the electron detectors. The M T and M B are scintillation 
counters, K T and K B are thin-plate spark chambers. Each electron detector 
consists of nine layers of lead, spark chamber, and scintillator sandwiched 

together. 



Figure 10 shows a calibration of one of the two electron detectors. The 
response of the telescope is plotted as a function of the energy of the beam. 
We see that the instrument is linear. The three sets of points in the upper 
curve correspond to measurements made at different times (given by the 
run number), and to two different positions of the beam in the detector. 
Near the extremes of the detector, the pulse-height decreases and the cali- 
brations are parametrized according to the maximum depth of the detector 
available to the shower. The two lower curves are two of these edge-effect 
calibrations. During the calibrations the detectors were rotated to many 
positions, and calibrated as a function of depth and energy in order to allow 
the calculation of the total efficiency for any event configuration. For fixed 
depths we see that the fluctuations are small; in any case the system was 
frequently calibrated in order to be sure that pions were not wrongly identi- 
fied as electrons. 

Figure 11a shows a complete efficiency calibration affixed energy. The 
purpose of this figure is to show that the electron detectors « top » and 
« bottom » had very similar characteristics. The open circles refer to the 
bottom detector and the full circles to the top detector. The electron energy 
for these two sets of points in 1050 Me V. Figure 116 shows a family of 
curves corresponding to 170 calibration points taken at energies from 1.05 GeV 
down to 0.45 GeV. 



The basic SU 3 mixing: oigi^Wj 



323 





Channels 
























Encoder -10 






















50 


















[ (e w , 


«D. 


Depth) RUNS 


48 
















* 








46 


- 














L{Ho. 


10 , 


49) 541-549 


44 














% 


/ 


;}(-io. 


10 


49) 582 - 592 


42 












\ 












40 










\ 


\ 






}[20. 


10 . 


49) 562-570 


38 














J 


j^io. 


15 , 


27) 551-559 


36 
34 
32 
30 
28 




* 


i 


i 


i 


i 




\ 


lit- 23, 

L 1 


16 , 


21) 573-580 


26 


- cot 




















24 


'///'^//'/i'/' 






















22 
























20 


paoortol 






















18 
























16 

























200 400 



600 



800 1000 1200 1300 

Energy of Electron 



Fig. 10. - Calibration of one of the electron detectors. Total scintillation coun- 
ter pulse-height vs. electron energy. 



Table I summarizes the efficiencies of the electron detectors. From as low as 
400 MeV up to 1 100 MeV, one can reject pions with a power of ~ 3 x 10~ 4 . For 
each particle and each momentum there are three numbers: the electronic 
efficiency, the picture analysis efficiency, and their product, the over-all 
efficiency. As mentioned above, the electron detectors consist of counters 
and spark chambers, so there is an electronic rejection in the trigger; then, 
once the pictures are taken, there is a further rejection in the picture analysis. 
The latter is very important because it allows the elimination of charge exchange 
of pions, which is the greatest source of trouble when you want to distinguish 



324 



A. Zichichi 




Fig. 11. - Efficiency calibration of the electron detector as a function of the 

depth available to a shower, a) Response of « top » (•) and « bottom » (o) 

counters at 1.05 GeV. b) Response of « bottom » counter at different energies. 

The curves correspond to 170 calibration points. 



a pion from an electron. From 400 MeV to 1100 MeV, the power of the 
telescope against pions is practically the same, and the efficiency for electron 
detection is very good — between 70 and 80%. 





Table I. - 


- Efficiency in the e 


lectron detector « bottc 


im ». 


Momentum 

(MeV/c) 


Par- 
ticle 


Electronic 
efficiency (%) 


Picture analysis 
efficiency (%) 


Over-all 
efficiency (%) 


400 
1100 


e 

7T 

e 

TV 


(77.5 ±2.2) 
( 6.3±0.2) 
(94.0 ±1.5) 
(17.6±0.6) 


(89.0 ±2.2 ) 
( 0.43±0.2 ) 
(88.0 ±2.0 ) 
( 0.16±0.16) 


(69.0±2.6) 

( 2.7±1.6)xl0" 4 

(83.0±2.3) 

( 2.8±2.8)X10~ 4 



To summarize, each of the two telescopes has ~ 3 x 10" 4 rejection against 
pions, giving a product ~ 10~ 7 , which is the rejection factor for charged nn 
pairs and any other sort of charged multipion events. It is this rejection power 
that allows the study of rare events such as (e+e") decays of strongly inter- 
acting particles. 

4'3. Some relevant details. - Table II summarizes the most relevant para- 
meters of the experiment for co and <j> decays into (e+e~). The co-mass region 
and the <j)-mass region have been investigated using the same experimental 



The basic SU S mixing: cog^co! 325 

set-up at different angles of acceptance for the neutron and electron detectors, 
but changing the primary beam momentum in order to maximize the number 
of observable events, i.e. (production cross-section) x (acceptance). For more 
details we refer the reader to the original papers (refs. [30] and [31]). 

Table II. - Relevant parameters of the to and <j> decay experiments (*). 



Parameter 


co (ref. [30]) 




4> (ref. [31]) 


Pn 




1.67 GeV/c 




1.93 GeV/c 


e i°ab 




31°^45° 




19°^33° 


e c.m. 




165°-^94° 




160°-^70° 


T a 




(42-^430) MeV 




(95^-560) MeV 


'« 




(46-M8) ns 




(32^17) ns 


a 2 




(0.08^0.8) (GeV/c) 


i 


(0.18-M.l) (GeV/c) 2 


AM 




± 10 MeV 




± 15 MeV 


G lab 




6°-^32° 




5°4-25° 


o» 




32° 




36° 


T x = 


i?! threshold 




1.7x(d£/dx) min 


2 T 


= 2-8 threshold 




150 MeV 




22 


threshold 




700 MeV 





(*) p n is the primary pion momentum. It has been chosen at the maximum of the production cross- 
section. 0^ is the angular range covered by the neutron detectors in the laboratory system. 6° m is the cor- 
responding value in the centre-of-mass system. T n is the range of neutron kinetic energies accepted in the 
above angular range. It follows the corresponding range of neutron time-of-flight / n . q* is the range of four- 
momentum transfer. AM is the mass uncertainty, dj'^ is the angular range of vector-meson production. 
6 D is the angular position of the electron detectors in the vertical plane containing the beam. 7\ and B x are 
the thresholds of the first counters in the electron detectors, i.e. after two radiation lengths in lead. S T and 
S B are the thresholds of the two electron detectors « top » and « bottom ». These thresholds were fixed at 
a very low value of 150 MeV incident electromagnetic energy in order to have high efficiency in the detection 
of electromagnetic showers. SS = X T + SB is the total electromagnetic energy released in «top» plus 
« bottom ». We trigger every time that the total energy is greater than 700 MeV; again this choice of low 
threshold is taken in order to have good detection efficiency for electromagnetic showers originated either by 
electrons or photons. 



Another important point worth mentioning is the way in which yy events 
are rejected. In the description of the electron detectors, it was pointed out 
that the rejection power against charged tz-k pairs was ~ 10~ 7 . But in (n~p) 
interactions, two or more 7i°'s can be produced; 7r°'s decay into y's, which 
then materialize in the target or in the plastic scintillator before the thin- 
plate spark chambers, thus producing electron-positron pairs which can 
simulate a genuine e ± from vector meson decay. It is possible to recognize 



326 



A. Zichichi 



most of these y-produced « fake e± », because they are really « electron- 
positron pairs » whose opening becomes sufficiently large by multiple scatter- 
ing in the trasversal of the material which is in front of the kinematic spark 
chambers. The distribution of the distance between two tracks of a pair is 
shown in Fig. 12. The wide part of the spectrum is that expected from 




5 10 15 20 

OISTANCE BETWEEN TWO TRACKS OF A PAIR IN mm 

Fig. 12. - Distribution of the distances in space between the two tracks of an 
electron-positron pair. 



multiple scattering. The peak at zero is clearly due to a genuine single e± 
and not (e+e~) pairs simulating single tracks. In fact, from the measured 
distribution (shown in Fig. 12) the number of « fake e± » present in the 



The basic SU 3 mixing: cog^Wj 



327 



peak is expected to be ~ 2. If we now plot the mass distribution of the events 
in the peak of Fig. 12, we obtain the distribution shown in Fig. 13. 

Notice that this result represents the first successful attempt to resolve 
the co-peak from the p. As mentioned in the introduction, the experimental 



N°ev/30MeV 



15~ 




I ' I ' I i T " | i I ' I " I i I ' | r | i | i I i i f | i i i i i 

700 800 900 MASS 



Fig. 13. - For to -> e+e~, mass distribution of e + e~ pairs in the co-region ob- 
tained with the same type of analysis as in the ^-case. The shape of the p-dis- 
tribution is determined by its natural width, the known production distribu- 
tion and density matrix, and the experimental acceptance and resolution. The 
dashed curve is the result of a maximum likelihood fit to the experimental data. 



328 



A. Zichichi 



conditions were chosen in such a way as to minimize the amount of observable 
p's. In fact, the broken curve is the expected p-mass distribution calculated 
from the known production and decay angular distribution combined with 
the experimental acceptance. 

Repeating the same type of analysis for the <j>-case gave the mass distri- 
bution shown in Fig. 14. In the cf>-mass region there is a total of ten events 




800 900 1000 noo 

MASS (MeV) 
Fig. 14. - For <j> -> e+e~, the mass distribution for those events with zero opening 

distance in Fig. 12. 



The basic SU 3 mixing: Wg^co! 329 

minus one background events. To have a small background was an essential 
feature of the experiment, the limitation in the number of observed <j> -> e+e - 
being due to the available machine time. Notice the difference between the 
distribution shown in Fig. 14 and that of the previous one shown in Fig. 13. 
The background below the <j)-peak is flat because there is no p-like object 
in the cj>-mass region. In conclusion, a total of nine events of unambiguously 
identified § -> e + e~ decays were observed. 

4*4. Results. - Table III summarize the experimental results obtained on 
the (e+e~) decay of co and c|> mesons. Let us start with the co column. The first 
entry is the direct experimental result obtained. Below there is the co-pro- 
duction cross-section which is well known; these two numbers then give the 
branching ratio, which together with the total width of the co, taken from the 
Rosenfeld tables [33], gives the partial width in the bottom entry. 

Table III. - Experimental results of <o and 9 decay. 

G)-»e+e- (ref. [30]) <j>->e+e- (ref. [31]) 

a(n-p -> nV) (67±25) x IO- 33 cm 2 (18.4±6.9) x 10- 33 cm 2 

l-> e+e - 

o(n-p -> nV) (1.67±0.07) x 10" 27 cm 2 (30±6) x lO" 30 cm 2 

Uall 

r(V^e+e-)/r(V->all) (4.0±1.5)xl0- 5 (6.1±2.6)xl0- 4 

T(V^ all) (*) (12.2±1.3) MeV (3.4±0.8) MeV 

r(V^e+e-) (0.49±1.19)keV (2.1±0.9) keV 

(*) Date taken from the Rosenfeld tables. 

In the case of the §, the production cross-section is much lower and it is 
not so well known as that of the co. In fact, a point in the ^-production 
cross-section was measured by the Bologna-CERN Collaboration [10], 
because when the experiment was started, the 4> production had not been 
observed in pion-nucleon interactions [34]. The energy at which the (^-pro- 
duction cross-section was measured is slightly higher than that at which the 
decay experiment was performed. In fact, a maximum value in the cross- 
section had still to be found, when a bubble chamber group [35] published 
a paper in which the maximum seemed to be 1 50 MeV lower ; so the experi- 
ment was performed at the lower beam momentum. Notice that the measured 
value of the (^-production cross-section [10] is in very good agreement with 



330 A. Zichichi 

the bubble chamber data [35]. Again the total width is taken from the 
Rosenfeld tables [33] to derive the partial width. 

The value of the generalized mixing angle 6 G was thus determined to be : 

tan0 G = , -> G = 23° 

This result is in excellent agreement with the « current mixing » theory of 
Kroll et al. [16] and of Oakes and Sakurai [18] (the slight difference between 
the KLZ and OS predictions for 6 G is due to the use of slightly different 
mass values, the two models of SU S symmetry breaking being identical [27]). 

In connection with previous remarks, the effect of (oo-p) interference has 
also been estimated [36], the result being a variation of ±3° for complete 
constructive or destructive interference respectively. It should be noticed, 
however, that in the OPE model the (o>p) interference is exactly zero. 

Following the theoretical considerations previously reviewed, the results 
obtained by the Bologna-CERN collaboration led to the following con- 
clusions [30, 31]: 

i) the general idea of (tOg-tO]) mixing is confirmed; 

ii) the First Generalized Weinberg Spectral Function Sum Rule (satu- 
rated using only p, to, (j>) is valid within 30 % over all experimental 
uncertainty; 

iii) there is no evidence for the coupling of the electromagnetic field 
to an SU S singlet; 

iv) the old A quantum number [37] is not a good quantum number; 

v) the fact that (e + e~) decays of co and § are observed with the meas- 
ured rates is a direct evidence that the J PG quantum numbers of 
the co and (j> are indeed 1 . 



5. - Present status and conclusions. 

The results obtained using strong production reactions (33) were followed 
by other measurements of Ting and collaborators at DESY [38] and later 
by the Orsay group [39]. 

Ting studied the photoproduction of p's and of rj>'s, thus obtaining the 
partial widths r p ^ e +e- and r^ e+e -. The DESY result with its uncertainties 
for 6q is shown in Fig. 15, where also the Orsay data are plotted. The Orsay 
group made use of the (e + e~) storage ring facility in order to determine the 



The basic SU Z mixing: cog^c^ 331 

partial widths r p ^ +e -, /^ e+e -, and r^+e-. As mentioned previously, the 
production reactions are given in eqs. (35)-(37), 



(35) 



(36) 



(37) 




e+e - -^ p -> Tu+Tr - 



e+e- -> to 



^7I°Y 



./ 



KK 



TC + 7T _ 7l 9 



where the vector mesons are produced « electromagnetically » and their 
strong decays are observed. The identification of the p, co, (j> is done using 
the information coming from the total (e+e~) energy, while the identification 
of the final states in the various reactions is performed via the use of geo- 
metrical constraints on the decay products in the various reactions (35)-(37). 

All experimental data available so far are reported in Fig. 15, where all 
the theoretical predictions are also shown. The diagram is constructed so 
as to reproduce in a graphically clear way the fst GWSR, as derived from 
DMO [17] and OS [18], i.e., relation (29). Notice that the quark model 
prediction [26] numerically satisfies the fst GWSR. This should not be so 
strange, as the results of the quark model can be derived from the following 
set of assumptions [40] : i) PCAC; ii) First and Second GWSR; iii) pole 
dominance in the First and Second GWSR. 

It should be emphasized that the two predictions for 6a, i-e., that of 
DMO [17] and that of KLZ [16] and OS [18], would coincide to first order 
SU S symmetry-breaking. These models are all of the « current mixing » type; 
they differ only by second-order SU 3 symmetry-breaking effects. Also the 
two « mass mixing » models of KLZ [16] differ only in second-order SU 3 
symmetry-breaking. 

It would be misleading to try combining the experimental data of Fig. 15 
in the hope of giving an answer to this extremely interesting question which 
refers to second-order SU 3 symmetry-breaking effects. There are, in fact, 
no other experiments where second-order SU 3 symmetry-breaking effects 
can so neatly be measured. 



332 



A. Zichichi 




V m <|> r (4>— e ++ 

Fig. 15. - Theoretical predictions and experimental measurements of the (co 8 -co x ) mixing. 



But we have extended the discussion too far. Let us not forget that a key- 
point in the great SU 3 castle has withstood the experimental proof: the reason 
why the unitary-symmetry mass formula of Gell-Mann and Okubo does 
not hold true for the vector meson multiplet is really the (cdg-o^) mixing 
mechanism. However, the field is now open for checking second-order 
effects in SU 3 symmetry-breaking. 



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List of Papers by Edoardo Amaldi 



1) Sulla dispersione anomala del mercurio e del litio (with E. Segre). 
Rend. Ace. Lincei, 1, 919-921 (1928). 

2) Sulla teoria dell'effetto Raman (with E. Segre). Rend. Ace. Lincei, 9, 
407-409 (1929). 

3) Sulla teoria quantistica deU'effetto Raman. Rend. Ace. Lincei, 9, 876- 
881 (1929). 

4) tiber die streuung von Roentgenstrahlen am Wasser. Phys. Z, 32, 
914-919 (1931). 

5) Sulla distribuzione delle molecole in un liquido. Riv. Nuovo Cimento, 
9, CXLI (1932). 

6) Uber den Ramaneffekt des CO. Z. Phys., 19, 492-494 (1932). 

7) Ramaneffekt des gasformingen Ammoniaks (with G. Placzek). Nature, 
20, 521 (1932) 

8) Spettri di assorbimento degli alcalini nel campo elettrico (with E. Segre). 
Ric. Scient., 42, 41-43 (1933). 

9) tiber das Ramanspektrum des gasformigen Ammoniaks (with G. 
Placzek). Z. Phys., 81, 259-269 (1933). 

10) Series of alkaline atoms in an electric field (with E. Segre). Nature, 
132, 444 (1933). 

11) Effetto del campo elettrico sul limite della serie del potassio (with E. 
Segre). Rend. Ace. Lincei, 19, 588-594 (1934). 

12) Effetto della pressione sui termini elevati degli alcalini (with E. Segre). 
Nuovo Cimento, 11, 145-156 (1934). 

13) Nuovi radioelementi prodotti con bombardamento di neutroni (with 
E. Fermi, F. Rasetti, E. Segre). Nuovo Cimento, 11, 442-451 (1934). 

14) Segno ed energia degli elettroni emessi da elementi attivati con neu- 
troni (with E. Segre). Nuovo Cimento, 11, 452-460 (1934). 

15) a) Effetto della pressione sui termini alti della serie degli alcalini (with 
E. Segre). Ric. Scient., 5, 53 (1934). 

b) Effect of pressure on high terms of alkaline spektra. Nature, 133, 
141 (1934). 



336 List of papers by Edoardo Amaldi 

16) Radioattivita provocata da bombardamento di neutroni. Ill (with 
O. D'Agostino. E. Fermi, F. Rasetti, E. Segre). Ric. Scient., 5, 452- 
453 (1934). 

17) Radioattivita provocata da bombardamento di neutroni. IV (with 
O. D'Agostino, E. Fermi, F. Rasetti, E. Segre). Ric. Scient., 5 X , 652- 
653 (1934). 

18) Radioattivita provocata da bombardamento di neutroni. V (with 
O. D'Agostino, E. Fermi, F. Rasetti, E. Segre). Ric. Scient., 5 2 , 21-22 
(1934). 

19) Azione di sostanze idrogenate sulla radioattivita provocata da neutroni 
(with E. Fermi, B. Pontecorvo, F. Rasetti, E. Segre). Ric. Scient., 5 2 , 
282-283 (1934). 

20) Radioattivita provocata da bombardamento di neutroni. VI (with 
O. D'Agostino, E. Segre). Ric. Scient., 5 2 , 381-382 (1934). 

21) Radioattivita provocata da bombardamento di neutroni. VII (with 
O. D'Agostino. E. Fermi, B. Pontecorvo, F. Rasetti, E. Segre). Ric. 
Scient., 5 2 , 467-470 (1934). 

22) Artificial radioactivity produced by neutron bombardament (with E. 
Fermi, O. D'Agostino, F. Rasetti. E. Segre). Proc. Royal Soc. {London), 
146, 483-500 (1934). 

23) Le orbite oos degli elementi (with E. Fermi). Mem. Ace. Italia, 6, 
119-149 (1934). 

24) Nuove radioattivita provocate da neutroni. La disintegrazione del boro. 
Nuovo Cimento, 12, 223-231 (1935). 

25) Radioattivita provocata da bombardamento di neutroni. VIII (with 
O. D'Agostino, E. Fermi, B. Pontecorvo, F. Rasetti, E. Segre). Ric. 
Scient., 6 l5 123-125 (1935). 

26) Radioattivita indotta da bombardamento di neutroni. IX (with O. D'Ago- 
stino, E. Fermi, B. Pontecorvo, E. Segre). Ric. Scient., 6 l5 435-437 
(1935). 

27) Radioattivita provocata da bombardamento di neutroni. X (with O. 
D'Agostino, E. Fermi, B. Pontecorvo, E. Segre). Ric. Scient., 6 l5 581- 
584 (1934). 

28) Sull'assorbimento dei neutroni lenti (with E. Fermi). Ric. Scient., 6 2 , 
334-347 (1935). 

29) Sull'assorbimento dei neutroni lenti. II (with E. Fermi). Ric. Scient., 
6 2 , 443-447 (1935). 

30) Artificial radioactivity produced by neutron bombardament. II (with 
O. D'Agostino, E. Fermi, B. Pontecorvo, R. Fasetti, E. Segre). Proc. 
Royal Soc. (London), 149 A, 522-558 (1935). 

31) Einige Spektroskopische Eigenschaften Hochangeregter Atome (with 
E. Segre). From the volume in honour of Zeeman, pp. 8-17 (1935). 



List of papers by Edoardo Amaldi 337 

32) Sull'assorbimento dei neutroni lenti. Ill (with E. Fermi). Ric. Scient., 
7 lt 56-59 (1936). 

33) Sul cammino libero medio dei neutroni lenti nella paraffina (with E. 

Fermi). Ric. Scient., 7 l9 223-225 (1936). 

34) Sui gruppi di neutroni lenti (with E. Fermi). Ric. Scient., 7 1? 310-313 
(1936). 

35) Sulle proprieta di diffusione dei neutroni lenti (with E. Fermi). Ric. 
Scient., 7 1? 393-395 (1936). 

36) a) Sopra l'assorbimento e la diffusione di neutroni lenti (with E. Fermi). 
Ric. Scient., l x , 454-503 (1936). 

b) On the absorption and diffusion of slow neutrons (with E. Fermi). 
Phys. Rev., 50, 899-926 (1936). 

37) Behaviour of slow neutrons at different temperatures (with E. Segre). 
Phys. Rev., 50, 571 (1936). 

38) Un generatore artificiale di neutroni (with E. Fermi, F. Rasetti). Ric. 
Scient., 8 2 , 40-43 (1937). 

39) Kunstliche Radioactivitat durch Neutronen. Phys. Z., 38, 692-734 (1937). 

40) Neutron yields from artificial sources (with L. R. Hafstad, N. A. Tuve). 
Phys. Rev., 51, 896-912 (1937). 

41) Metodo fotografico per il rilievo della fluttuazione dei raggi X emessi 
da un'ampolla (with G. C. Trabacchi). Rend. 1st. Sanita, 1, 317-320 
(1937). 

42) Sulle quantita di sostanze radioattive artificiali che si possono prepa- 
rare con diversi processi. Atti SIPS (1938). 

43) On the albedo of slow neutrons (with E. Fermi, G. C. Wick). Phys. 
Rev., 53, 493 (1938). 

44) Sopra la conversione interna dei raggi gamma e X del RaD (with F. 
Rasetti). Ric. Scient., 10, 111-114 (1939). 

45) Sulle radiazioni emesse dal gadolinio per cattura dei neutroni lenti 
(with F. Rasetti). Ric. Scient., 10, 115-118 (1939). 

46) Generatore di neutroni a 1.000 kV (with D. Bocciarelli, F. Rasetti, 
G. C. Trabacchi). Ric. Scient., 10, 623-632 (1939). 

47) Sulla diffusione dei neutroni prodotti nella reazione: 12 C+ 2 D = 13 N+ X n 
(with D. Bocciarelli, F. Rasetti, G. C. Trabacchi). Ric. Scient., 10, 
633-637 (1939). 

48) On the scattering of neutrons from the C+D reaction (with D. Boc- 
ciarelli, F. Rasetti, G. C. Trabacchi). Phys. Rev., 56, 881-884 (1939). 

49) Misura della sezione d'urto elastico fra neutroni e protoni (with D. Boc- 
ciarelli, G. C. Trabacchi). 

a) Atti Ace. d' Italia, 1, 350-358 (1940). 

b) Ric. Scient., 11, 121-127 (1940). 



22 



338 List of papers by Edoardo Amaldi 

50) Sulla scissione degli elementi pesanti (with M. Ageno, D. Bocciarelli, 
B. N. Cacciapuoti, G. C. Trabacchi). 

a) Atti Ace. <T Italia, 1, 525-536 (1940). 

b) Ric. Scient., 11, 302-311 (1940). 

51) Sulla scissione dell'uranio con neutroni veloci (with M. Ageno, D. Boc- 
ciarelli, G. C. Trabacchi). 

a) Atti Ace. <T Italia, 1, 746-751 (1940). 

b) Ric. Scient., 11, 413-417 (1940). 

52) Sulla distribuzione angolare dei neutroni emessi nella disintegrazione 
di elementi leggeri (with M. Ageno, D. Bocciarelli, G. C. Trabacchi). 
Rend. Ace. d* Italia, 2, 338-350 (1940). 

53) L'impianto generatore di neutroni a 1.000 kilowat dell'Istituto di Sanita 
Pubblica (with M. Ageno, D. Bocciarelli, G. C. Trabacchi). Rend. 1st. 
Sanita, 3, 201-216 (1940). 

54) II sistema neutrone-protone. Mem. Ace. d'ltalia, 13, 555-615 (1941). 

55) Sulla scissione del torio e del protoattinio (with D. Bocciarelli, G. C. 
Trabacchi). 

a) Ric. Scient., 12, 134-138 (1941). 

b) Rend. 1st. Sanita, 4, 266-272 (1941). 

56) Distribuzione angolare di raggi y emessi nella reazione: 1 H+ 7 Li = 
= 8 Be+y (with M. Ageno, D. Bocciarelli, G. C. Trabacchi). 

a) Ric. Scient., 12, 139-143 (1941). 

b) Rend. 1st. Sanita, 4, 260-265 (1941). 

57) Sull'urto fra protoni e neutroni. I (with M. Ageno, D. Bocciarelli, 
G. C. Trabacchi). Ric. Scient., 12, 830-842 (1941). 

58) Fission yield by fast neutrons (with M. Ageno, D. Bocciarelli, B. N. 
Cacciapuoti, G. C. Trabacchi). Phys. Rev., 60, 67-75 (1941). 

59) Sull'urto fra protoni e neutroni. II (with D. Bocciarelli, B. Ferretti, 
G. C. Trabacchi). Ric. Scient., 13, 502-531 (1942). 

60) Tubo a raggi X con catodo sostituibile (with D. Bocciarelli, G. C. Tra- 
bacchi). Rend. 1st. Sanita, 5, 694-698 (1942). 

61) Streuung von 14-MV-Neutronen an Protonen (with D. Bocciarelli, 
B. Ferretti, G. C. Trabacchi). Naturwiss., 30, 582-583 (1942). 

62) Sull'urto di neutroni contro protoni e deutoni (with M. Ageno, D. Boc- 
ciarelli, G. C. Trabacchi). 

a) Nuovo Cimento, 1, 253-278 (1943). 

b) Gazz. Chim. Ital, 75 (1943). 

63) Apparecchio per la purificazione e la conservazione di idrogeno desti- 
nato a produzione di ioni (with D. Bocciarelli, G. C. Trabacchi). 

a) Rend. 1st. Sanita, 6, 416-419 (1943). 

b) Gazz. di Chim., 74, 127-130 (1944). 

64) Streuung von schnellen Neutronen an Protonen und Deuteronen (with 
M. Ageno, D. Bocciarelli, B. N. Cacciapuoti, G. C. Trabacchi). Natur- 
wiss., 31, 231-232 (1943). 



List of papers by Edoardo Amaldi 339 

65) Effetti di diffrazione nello sparpagliamento dei neutroni veloci (with 
D. Bocciarelli, B. N. Cacciapuoti, G. C. Trabacchi). 

a) Atti Ace. Lincei, 1, 29-34 (1946). 

b) Rend. 1st. Sanitd, 9, 5-13 (1946). 

66) Contribute* alia teoria dell'acceleratore a induzione (with B. Ferretti). 
Atti Ace. Lincei, 1, 85-89 (1946). 

67) Sue due varianti dell'acceleratore a induzione (with B. Ferretti). Nuovo 
Cimento, 3, 22-39 (1946). 

68) Sullo sparpagliamento elastico dei neutroni veloci da parte di nuclei 
medi e pesanti (with D. Bocciarelli, B. N. Cacciapuoti, G. C. Trabacchi). 

a) Nuovo Cimento, 3, 203-234 (1946). 

b) Rend. 1st. Sanitd, 9, 687-724 (1946). 

69) On two possible modifications of the induction accelerator (with B. 
Ferretti). Rev. Scient. Instr., 17, 389-395 (1946). 

70) On the scattering of fast neutrons by protons and deuterons (with 
M. Ageno, D. Bocciarelli, G. C. Trabacchi). Phys. Rev., 71, 30-31 (1947). 

71) a) Sulla dipendenza del raggio nucleare dal peso atomico (with B. N. 
Cacciapuoti). Atti Ace. Lincei. 2, 243-246 (1947). 

b) On the dependence of nuclear radius on the mass number (with 
B. N. Cacciapuoti). Phys. Rev., 71, 739-740 (1947). 

72) The elastic scattering of fast neutrons by medium and heavy nuclei 
(with D. Bocciarelli, B. N. Cacciapuoti, G. C. Trabacchi). Phys. Soc. 
Cambr. Conf. Report, 97-113 (1947). 

73) A research for anomalous scattering of [x-mesons by nucleons (with 
G. Fidecaro). Helv. Phys. Acta, 23, 93-102 (1950). 

74) Contributo alio studio degli sciami estesi. I (with C. Castagnoli, A. Gigli, 
S. Sciuti). Nuovo Cimento, 7, 401-456 (1950). 

75) An experiment on the anomalous scattering of fast [x-mesons by nucleons 
(with G. Fidecaro). Nuovo Cimento, 7, 535-552 (1950). 

76) On the Coulomb scattering of (x-mesons by light nuclei (with G. Fide- 
caro, F. Mariani). Nuovo Cimento, 7, 553-574 (1950). 

77) Sull'effetto di transizione nel fenomeno di produzione di stelle da parte 
della radiazione cosmica (with C. Castagnoli, A. Gigli, S. Sciuti). Nuovo 
Cimento, 7, 697-699 (1950). 

78) On the influence of the spin interaction on the scattering of mesons 
and electrons by licht nuclei (with G. Fidecaro, F. Mariani). Nuovo 
Cimento, 7, 757-773 (1950). 

79) Contributo alio studio degli sciami estesi. II (with C. Castagnoli, A. 
Gigli, S. Sciuti). Nuovo Cimento, 7, 816-834 (1950). 

80) An experiment on the anomalous scattering of ^-mesons by nucleons 
(with G. Fidecaro). Phys. Rev., 81, 338-341 (1951). 



340 List of papers by Edoardo Amaldi 

81) Diffraction effects in the scattering of neutrons, mesons, and electrons 
by nuclei. Ripon Professorship Lecture, 1951, delivered in the Indian 
Associations for the Cultivation of Science, Jadapur, Calcutta 32, in 
January 1951. 

82) On the interaction of cosmic rays with matter under 50 metres water 
equivalent (with C. Castagnoli, A. Gigli, S. Sciuti). Nuovo Cimento, 9, 
453-455 (1952). 

83) An anticoincidence experiment on cosmic rays at a depth of 50 metres 
water equivalent (with C. Castagnoli, S. Sciuti, A. Gigli). Proc. Phys. 
Soc., A 65, 556-558 (1952). 

84) On the interaction of cosmic rays with matter under 50 metres water 
equivalent. II (with C. Castagnoli, A. Gigli, S. Sciuti). Nuovo Cimento, 
9, 969-1003 (1952). 

85) On the longitudinal development of air showers according to Fermi's 
theory of meson production (with L. Mezzetti, G. Stoppini). Nuovo 
Cimento, 10, 803-816 (1953). 

86) Underground experiments in Europe. Proc. Intern. Conf. Theoretical 
Physics (Kyoto-Tokyo), September 1953. 

87) Contribution to the tau-meson investigation (with G. Baroni, C. Casta- 
gnoli, G. Cortini, A. Manfredini). Nuovo Cimento, 10, 937-948 (1953). 

88) Preliminary research on V\ events in emulsion (with C. Castagnoli, 

G. Cortini, A. Manfredini). Nuovo Cimento, 10, 1351-1353 (1953). 

89) On a possible negative K -> rc-meson decay (with G. Baroni, C. Casta- 
gnoli, G. Cortini, C. Franzinetti, A. Manfredini). Nuovo Cimento, 11, 
207-209 (1954). 

90) Lifetimes measurement of unstable charged particles of cosmic radia- 
tion using emulsion (with C. Castagnoli, G. Cortini, C. Franzinetti). 
Nuovo Cimento, 12, 668-676 (1954). 

91) On the interaction of fast fx-mesons with matter. Suppl. al Nuovo Cimento, 
11, 406-413 (1954). 

92) Contribution to the jji-meson investigation (with G. Baroni, G. Cortini, 
C. Franzinetti). Suppl. al Nuovo Cimento, 12, 181-184 (1954). 

93) Contribution to the K-meson investigation (with G. Cortini, A. Man- 
fredini). Suppl. al Nuovo Cimento, 12, 205-206 (1954). 

94) On the longitudinal development of air showers according to Fermi's 
Theory of meson production (with G. Stoppini). Memorie del V Con- 
gresso Inter nazionale de Radiation Cosmica, Mexico 1958. 

95) Unusual event produced by cosmic ray (with C. Castagnoli, G. Cortini, 
C. Franzinetti, A. Manfredini). Nuovo Cimento, 1, 492-500 (1955). 

96) On the measurement of the mean lifetime of strange particles. Suppl. 
al Nuovo Cimento, 2, 253 (1955). 



List of papers by Edoardo Amaldi 341 

97) a) Su di una Stella provocata da un antiprotone osservato in emul- 
sioni nucleari (with. O. Chamberlain, W. Chupp, G. Goldhaber, E. 
Segre, C. Wiegand, G. Baroni, C. Castagnoli, C. Franzinetti, A. Man- 
fredini). Rend. Ace. Lincei, Ser. VIII, 19, 381 (1955). 

b) Antiproton star observed in emulsion (with O. Chamberlain, W. 
Chupp, G. Goldhaber, E. Segre, C. Wiegand, G. Baroni, C. Castagnoli, 
C. Franzinetti, A. Manfredini). Phys. Rev., 101, 909 (1956). 

98) On the observation of an antiproton star in emulsion exposed at the 
Bevatron (with O. Chamberlain, W. Chupp, G. Goldhaber, E. Segre, 
C. Wiegand, G. Baroni, C. Castagnoli). Nuovo Cimento, 3, 447 (1956). 

99) Example of an anti-nucleon annihilation (with O. Chamberlain, W. 
Chupp, A. G. Ekspong, G. Goldhaber, S. Goldhaber, E. J. Lofgren, 
C. Wiegand, G. Baroni, C. Castagnoli, C. Franzinetti, A. Manfredini). 
Phys. Rev., 102, 921 (1956). 

100) An electronic scanner for nuclear emulsion (with C. Castagnoli C. 
Franzinetti). Nuovo Cimento, 4, 1165-1173 (1956). 

101) Report on the T-mesons. Nuovo Cimento Suppl. al Vol. 4, n. 2 179-215 
(1956). 

102) Antiproton-nucleon annihilation process (antiproton collaboration 
experiment) (with W. H. Barkas, R. W. Birge, W. W. Chupp, A. 
G. Ekspong, G. Goldhaber, S. Goldhaber, H. H. Heckman, D. H. 
Perkins, J. Sanweiss, E. Segre, F. M. Smith, D. H. Stork, L. van 
Rossum). Phys. Rev., 105, 3, 1037-1058 (1957). 

103) Further results on antiproton annihilations (with C. Castagnoli, M. 
Ferro-Luzzi, C. Franzinetti, A. Manfredini). Nuovo Cimento, 5, 1797- 
1800 (1957). 

104) The production and slowing down of neutrons. Handbuck der Physik, 
Vol. XXXVIII/2 pp. 1-659 (Springer Verlag, Berlin, 1959). 

105) Study of antiproton with emulsion technique (with G. Baroni, G. 
Bellettini, C. Castagnoli, M. Ferro-Luzzi, A. Manfredini). Nuovo 
Cimento, 14, 977-1026 (1959). 

106) Production and decay of an antisigma+ (with A. Barbaro-Galtieri, 
G. Baroni, C. Castagnoli, M. Ferro-Luzzi, A. Manfredini, M, Muchinik, 
V. Rossi e M. Severi). Nuovo Cimento, 16, 392-395 (1960). 

107) Le antiparticelle. Suppl. al Nuovo Cimento, 19, 101 (1961). 

108) Search for Dirac magnetic poles (with G. Baroni, H. Bradner, H. G. de 
Carvalho, L. Hoffmann, A. Manfredini e G. Vanderhaeghe). Comptes 
Rendus de la Conf. Intern. d'Aix-en-Provence, sur le Particules elemen- 
taires, 155(1961). 

109) Experimental data on spectral variations during forbush (with F. Ba- 
chelet, P. Balata, N. Iucci). Pontificiae Academiae Scientiarum Scripta 
Varia. Semaine d'Etude dur le probleme du Rayonnement cosmique 
dans l'espace interplanetaire 1-6 October 1962, p. 299. 



342 List of papers by Edoardo Amaldi 

110) Search for Dirac magnetic poles (with G. Baroni, A. Manfredini, H. 
Bradner, L. Hoffmann, G. Vanderaeghe). Nuovo Cimento, 28, 773 (1963). 

111) Search for Dirac magnetic poles (wi h G. Baroni, H. Bradner, H. G. de 
Carvalho, L. Hoffmann, A. Manfredini e G. Vanderaeghe). CERN- 63-13 
Nuclear Physics Division, 10th April 1963. 

1 12) Polarization of recoil protons in e-p elastic scattering (with E. Gans- 
sauge, R. Gomez, G. Gorini, S. Penner, S. Serbassi, G. Stoppini, M. 
L. Vincelli). Nuovo Cimento, 39, 474 (9165). 

113) Experimental search for a possible change of the beta decay constant 
with centrifugae forces (with M. Ageno). Atti Ace. Lincei, Memorie, 8, 
Sez. II, fase I (1966). 

114) Realta naturale e teorie scientifiche, in Saggi su Galileo, 1967. 

115) On the Dirac magnetic poles, in Old and new Problems in Elementary 
Particles edited by G. Puppi (Academic Press, New York, 1968). 

116) Ricordo di Ettore Majorana. Giornale di Fisica, 9, n. 4 (1968). 

117) L' opera scientifica di Ettore Majorana. Phyxics, 9, 3 (1968). See also 
La vita e V opera di Ettore Majorana (1906-1938). Accademia Nazionale 
dei Lincei (1966). 

118) A measurement of pion electroproduction cross-section near threshold 
(with M. Balla, B. Borgia, G. V. Di Giorgio, A. Giazotto, M. Giorgi, 
P. Pistilli, S. Serbassi, G. Stoppini). Lett. Nuovo Cimento, 1, 247 (1969). 



Not included in the list, a number of books of physics for high school, written in co- 
operation with Ginestra Amaldi. 



Tipografia Compositori - Bologna - Italy